Ellipsoid-Auswahl angepasst

revert Delete Tests/gha_resultsKarney.pkl

1,lambda-ES-CMA.py

@@ -1,364 +0,0 @@
-import numpy as np
-import plotly.graph_objects as go
-from ellipsoide import EllipsoidTriaxial
-
-
-def ell2cart(ell: EllipsoidTriaxial, beta, lamb):
-    x = ell.ax * np.cos(beta) * np.cos(lamb)
-    y = ell.ay * np.cos(beta) * np.sin(lamb)
-    z = ell.b  * np.sin(beta)
-    return np.array([x, y, z], dtype=float)
-
-
-def liouville(ell: EllipsoidTriaxial, beta, lamb, alpha):
-    k = (ell.Ee**2) / (ell.Ex**2)
-    c_sq = (np.cos(beta)**2 + k * np.sin(beta)**2) * np.sin(alpha)**2 \
-           + k * np.cos(lamb)**2 * np.cos(alpha)**2
-    return c_sq
-
-
-def normalize_angles(beta, lamb):
-    beta = np.clip(beta, -np.pi / 2.0, np.pi / 2.0)
-    lamb = (lamb + np.pi) % (2 * np.pi) - np.pi
-    return beta, lamb
-
-
-def compute_azimuth(beta0: float, lamb0: float,
-                    beta1: float, lamb1: float) -> float:
-    dlam = lamb1 - lamb0
-    y = np.cos(beta1) * np.sin(dlam)
-    x = np.cos(beta0) * np.sin(beta1) - np.sin(beta0) * np.cos(beta1) * np.cos(dlam)
-    alpha = np.arctan2(y, x)
-    return alpha
-
-
-def sphere_forward_step(beta0: float,
-                        lamb0: float,
-                        alpha0: float,
-                        s: float,
-                        R: float) -> tuple[float, float]:
-
-    delta = s / R
-    sin_beta0 = np.sin(beta0)
-    cos_beta0 = np.cos(beta0)
-
-    sin_beta2 = sin_beta0 * np.cos(delta) + cos_beta0 * np.sin(delta) * np.cos(alpha0)
-    beta2 = np.arcsin(sin_beta2)
-
-    y = np.sin(alpha0) * np.sin(delta) * cos_beta0
-    x = np.cos(delta) - sin_beta0 * sin_beta2
-    dlam = np.arctan2(y, x)
-    lamb2 = lamb0 + dlam
-
-    beta2, lamb2 = normalize_angles(beta2, lamb2)
-    return beta2, lamb2
-
-
-def local_step_objective(candidate: np.ndarray,
-                         beta_start: float,
-                         lamb_start: float,
-                         alpha_prev: float,
-                         c_target: float,
-                         step_length: float,
-                         ell: EllipsoidTriaxial,
-                         beta_pred: float,
-                         lamb_pred: float,
-                         w_L: float = 5.0,
-                         w_d: float = 5.0,
-                         w_p: float = 2.0,
-                         w_a: float = 2.0) -> float:
-
-    beta1, lamb1 = candidate
-    beta1, lamb1 = normalize_angles(beta1, lamb1)
-
-    P0 = ell2cart(ell, beta_start, lamb_start)
-    P1 = ell2cart(ell, beta1, lamb1)
-    d = np.linalg.norm(P1 - P0)
-
-    alpha1 = compute_azimuth(beta_start, lamb_start, beta1, lamb1)
-
-    c1 = liouville(ell, beta1, lamb1, alpha1)
-    J_L = (c1 - c_target) ** 2
-
-    J_d = (d - step_length) ** 2
-
-    d_beta = beta1 - beta_pred
-    d_lamb = lamb1 - lamb_pred
-    d_ang2 = d_beta**2 + (np.cos(beta_pred) * d_lamb)**2
-    J_p = d_ang2
-
-    d_alpha = np.arctan2(np.sin(alpha1 - alpha_prev),
-                         np.cos(alpha1 - alpha_prev))
-    J_a = d_alpha**2
-
-    return w_L * J_L + w_d * J_d + w_p * J_p + w_a * J_a
-
-
-
-
-def ES_CMA_step(beta_start: float,
-                lamb_start: float,
-                alpha_prev: float,
-                c_target: float,
-                step_length: float,
-                ell: EllipsoidTriaxial,
-                beta_pred: float,
-                lamb_pred: float,
-                sigma0: float,
-                stopfitness: float = 1e-18) -> tuple[float, float]:
-
-    N = 2
-    xmean = np.array([beta_pred, lamb_pred], dtype=float)
-    sigma = sigma0
-    stopeval = int(400 * N**2)
-
-    lamb = 30
-    mu = 1
-    weights = np.array([1.0])
-    mueff = 1.0
-
-    cc = (4 + mueff / N) / (N + 4 + 2 * mueff / N)
-    cs = (mueff + 2) / (N + mueff + 5)
-    c1 = 2 / ((N + 1.3)**2 + mueff)
-    cmu = min(1 - c1,
-              2 * (mueff - 2 + 1/mueff) / ((N + 2)**2 + mueff))
-    damps = 1 + 2 * max(0, np.sqrt((mueff - 1) / (N + 1)) - 1) + cs
-
-    pc = np.zeros(N)
-    ps = np.zeros(N)
-    B = np.eye(N)
-    D = np.eye(N)
-    C = B @ D @ (B @ D).T
-    eigeneval = 0
-    chiN = np.sqrt(N) * (1 - 1 / (4 * N) + 1 / (21 * N**2))
-
-    counteval = 0
-    arx = np.zeros((N, lamb))
-    arz = np.zeros((N, lamb))
-    arfitness = np.zeros(lamb)
-
-    while counteval < stopeval:
-        for k in range(lamb):
-            arz[:, k] = np.random.randn(N)
-            arx[:, k] = xmean + sigma * (B @ D @ arz[:, k])
-
-            arfitness[k] = local_step_objective(
-                arx[:, k],
-                beta_start, lamb_start,
-                alpha_prev,
-                c_target,
-                step_length,
-                ell,
-                beta_pred, lamb_pred
-            )
-            counteval += 1
-
-        idx = np.argsort(arfitness)
-        arfitness = arfitness[idx]
-        arindex = idx
-
-        xold = xmean.copy()
-        xmean = arx[:, arindex[:mu]] @ weights
-        zmean = arz[:, arindex[:mu]] @ weights
-
-        ps = (1 - cs) * ps + np.sqrt(cs * (2 - cs) * mueff) * (B @ zmean)
-        norm_ps = np.linalg.norm(ps)
-        hsig = norm_ps / np.sqrt(1 - (1 - cs)**(2 * counteval / lamb)) / chiN < 1.4 + 2 / (N + 1)
-        hsig = 1.0 if hsig else 0.0
-        pc = (1 - cc) * pc + hsig * np.sqrt(cc * (2 - cc) * mueff) * (B @ D @ zmean)
-
-        BDz = B @ D @ arz[:, arindex[:mu]]
-        C = (1 - c1 - cmu) * C \
-            + c1 * (np.outer(pc, pc) + (1 - hsig) * cc * (2 - cc) * C) \
-            + cmu * BDz @ np.diag(weights) @ BDz.T
-
-        sigma = sigma * np.exp((cs / damps) * (norm_ps / chiN - 1))
-
-        if counteval - eigeneval > lamb / ((c1 + cmu) * N * 10):
-            eigeneval = counteval
-            C = (C + C.T) / 2.0
-            eigvals, B = np.linalg.eigh(C)
-            D = np.diag(np.sqrt(np.maximum(eigvals, 1e-20)))
-
-        if arfitness[0] <= stopfitness:
-            break
-
-    xmin = arx[:, arindex[0]]
-    beta1, lamb1 = normalize_angles(xmin[0], xmin[1])
-    return beta1, lamb1
-
-
-def march_geodesic(beta1: float,
-                   lamb1: float,
-                   alpha1: float,
-                   S_total: float,
-                   step_length: float,
-                   ell: EllipsoidTriaxial):
-
-    beta_curr = beta1
-    lamb_curr = lamb1
-    alpha_curr = alpha1
-
-    betas = [beta_curr]
-    lambs = [lamb_curr]
-    alphas = [alpha_curr]
-
-    c_target = liouville(ell, beta_curr, lamb_curr, alpha_curr)
-
-    total_distance = 0.0
-    R_sphere = ell.ax
-    sigma0 = 1e-10
-
-    while total_distance < S_total - 1e-6:
-        remaining = S_total - total_distance
-        this_step = min(step_length, remaining)
-
-        beta_pred, lamb_pred = sphere_forward_step(
-            beta_curr, lamb_curr, alpha_curr,
-            this_step, R_sphere
-        )
-
-        beta_next, lamb_next = ES_CMA_step(
-            beta_curr, lamb_curr, alpha_curr,
-            c_target,
-            this_step,
-            ell,
-            beta_pred,
-            lamb_pred,
-            sigma0=sigma0
-        )
-
-        P_curr = ell2cart(ell, beta_curr, lamb_curr)
-        P_next = ell2cart(ell, beta_next, lamb_next)
-        d_step = np.linalg.norm(P_next - P_curr)
-        total_distance += d_step
-
-        alpha_next = compute_azimuth(beta_curr, lamb_curr, beta_next, lamb_next)
-
-        beta_curr, lamb_curr, alpha_curr = beta_next, lamb_next, alpha_next
-        betas.append(beta_curr)
-        lambs.append(lamb_curr)
-        alphas.append(alpha_curr)
-
-    return np.array(betas), np.array(lambs), np.array(alphas), total_distance
-
-
-
-if __name__ == "__main__":
-    ell = EllipsoidTriaxial.init_name("BursaSima1980round")
-
-    beta1 = np.deg2rad(10.0)
-    lamb1 = np.deg2rad(10.0)
-    alpha1 = 0.7494562507041596         # Ergebnis 20°, 20°
-
-    STEP_LENGTH = 500.0
-    S_total = 1542703.1458877102
-
-    betas, lambs, alphas, S_real = march_geodesic(
-        beta1, lamb1, alpha1,
-        S_total,
-        STEP_LENGTH,
-        ell
-    )
-
-    print("Anzahl Schritte:", len(betas) - 1)
-    print("Resultierende Gesamtstrecke (Chord, Ellipsoid):", S_real, "m")
-    print("Letzter Punkt (beta, lambda) in Grad:",
-          np.rad2deg(betas[-1]), np.rad2deg(lambs[-1]))
-
-def plot_geodesic_3d(ell: EllipsoidTriaxial,
-                     betas: np.ndarray,
-                     lambs: np.ndarray,
-                     n_beta: int = 60,
-                     n_lamb: int = 120):
-
-    beta_grid = np.linspace(-np.pi/2, np.pi/2, n_beta)
-    lamb_grid = np.linspace(-np.pi, np.pi, n_lamb)
-    B, L = np.meshgrid(beta_grid, lamb_grid, indexing="ij")
-
-    Xs = ell.ax * np.cos(B) * np.cos(L)
-    Ys = ell.ay * np.cos(B) * np.sin(L)
-    Zs = ell.b  * np.sin(B)
-
-    Xp = ell.ax * np.cos(betas) * np.cos(lambs)
-    Yp = ell.ay * np.cos(betas) * np.sin(lambs)
-    Zp = ell.b  * np.sin(betas)
-
-    fig = go.Figure()
-
-    fig.add_trace(
-        go.Surface(
-            x=Xs,
-            y=Ys,
-            z=Zs,
-            opacity=0.6,
-            showscale=False,
-            colorscale="Viridis",
-            name="Ellipsoid"
-        )
-    )
-
-    fig.add_trace(
-        go.Scatter3d(
-            x=Xp,
-            y=Yp,
-            z=Zp,
-            mode="lines+markers",
-            line=dict(width=5, color="red"),
-            marker=dict(size=4, color="black"),
-            name="ES-CMA Schritte"
-        )
-    )
-
-    fig.add_trace(
-        go.Scatter3d(
-            x=[Xp[0]], y=[Yp[0]], z=[Zp[0]],
-            mode="markers",
-            marker=dict(size=6, color="green"),
-            name="Start"
-        )
-    )
-    fig.add_trace(
-        go.Scatter3d(
-            x=[Xp[-1]], y=[Yp[-1]], z=[Zp[-1]],
-            mode="markers",
-            marker=dict(size=6, color="blue"),
-            name="Ende"
-        )
-    )
-
-    fig.update_layout(
-        title="ES-CMA",
-        scene=dict(
-            xaxis_title="X [m]",
-            yaxis_title="Y [m]",
-            zaxis_title="Z [m]",
-            aspectmode="data"
-        )
-    )
-
-    fig.show()
-
-plot_geodesic_3d(ell, betas, lambs)
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-

ES/Hansen_ES_CMA.py

@@ -1,7 +1,8 @@
 import numpy as np
+from numpy.typing import NDArray
 
 
-def felli(x):
+def felli(x: NDArray) -> float:
     N = x.shape[0]
     if N < 2:
         raise ValueError("dimension must be greater than one")
@@ -9,10 +10,9 @@ def felli(x):
     return float(np.sum((1e6 ** exponents) * (x ** 2)))
 
 
-def escma(func, *, N=10, xmean=None, sigma=0.5, stopfitness=1e-10, stopeval=None,
-          func_args=(), func_kwargs=None, seed=None,
-          bestEver = np.inf, noImproveGen = 0, absTolImprove = 1e-12, maxNoImproveGen = 100, sigmaImprove = 1e-12):
-
+def escma(func, *, N=10, xmean=None, sigma=0.5, stopfitness=1e-14, stopeval=2000,
+          func_args=(), func_kwargs=None, seed=0,
+          bestEver=np.inf, noImproveGen=0, absTolImprove=1e-12, maxNoImproveGen=100, sigmaImprove=1e-12):
     if func_kwargs is None:
         func_kwargs = {}
 
@@ -27,7 +27,7 @@ def escma(func, *, N=10, xmean=None, sigma=0.5, stopfitness=1e-10, stopeval=None
         N = xmean.shape[0]
 
     if stopeval is None:
-        stopeval = int(1e3 * N**2)
+        stopeval = int(1e3 * N ** 2)
 
     # Strategy parameter setting: Selection
     lambda_ = 4 + int(np.floor(3 * np.log(N)))
@@ -37,14 +37,14 @@ def escma(func, *, N=10, xmean=None, sigma=0.5, stopfitness=1e-10, stopeval=None
     weights = np.log(mu + 0.5) - np.log(np.arange(1, int(mu) + 1))
     mu = int(np.floor(mu))
     weights = weights / np.sum(weights)
-    mueff = np.sum(weights)**2 / np.sum(weights**2)
+    mueff = np.sum(weights) ** 2 / np.sum(weights ** 2)
 
     # Strategy parameter setting: Adaptation
     cc = (4 + mueff / N) / (N + 4 + 2 * mueff / N)
     cs = (mueff + 2) / (N + mueff + 5)
-    c1 = 2 / ((N + 1.3)**2 + mueff)
+    c1 = 2 / ((N + 1.3) ** 2 + mueff)
     cmu = min(1 - c1,
-              2 * (mueff - 2 + 1 / mueff) / ((N + 2)**2 + 2 * mueff))
+              2 * (mueff - 2 + 1 / mueff) / ((N + 2) ** 2 + 2 * mueff))
     damps = 1 + 2 * max(0, np.sqrt((mueff - 1) / (N + 1)) - 1) + cs
 
     # Initialize dynamic (internal) strategy parameters and constants
@@ -54,7 +54,7 @@ def escma(func, *, N=10, xmean=None, sigma=0.5, stopfitness=1e-10, stopeval=None
     D = np.eye(N)
     C = B @ D @ (B @ D).T
     eigeneval = 0
-    chiN = np.sqrt(N) * (1 - 1/(4*N) + 1/(21 * N**2))
+    chiN = np.sqrt(N) * (1 - 1 / (4 * N) + 1 / (21 * N ** 2))
 
     # Generation Loop
     counteval = 0
@@ -64,7 +64,7 @@ def escma(func, *, N=10, xmean=None, sigma=0.5, stopfitness=1e-10, stopeval=None
 
     gen = 0
 
-    print(f'    [CMA-ES] Start: lambda = {lambda_}, sigma ={round(sigma, 6)}, stopeval = {stopeval}')
+    # print(f'    [CMA-ES] Start: lambda = {lambda_}, sigma ={round(sigma, 6)}, stopeval = {stopeval}')
 
     while counteval < stopeval:
         gen += 1
@@ -91,27 +91,24 @@ def escma(func, *, N=10, xmean=None, sigma=0.5, stopfitness=1e-10, stopeval=None
             bestEver = fbest
             noImproveGen = 0
         else:
-            noImproveGen = noImproveGen + 1
+            noImproveGen += 1
 
-
-        if gen == 1 or gen%50==0:
-            print(f'    [CMA-ES] Gen {gen}, best = {round(fbest, 6)}, sigma = {sigma:.3g}')
+        if gen == 1 or gen % 50 == 0:
+            # print(f'    [CMA-ES] Gen {gen}, best = {round(fbest, 6)}, sigma = {sigma:.3g}')
+            pass
 
         if noImproveGen >= maxNoImproveGen:
-            print(f'    [CMA-ES] Abbruch: keine Verbesserung > {round(absTolImprove, 3)} in {maxNoImproveGen} Generationen.')
+            # print(f'    [CMA-ES] Abbruch: keine Verbesserung > {round(absTolImprove, 3)} in {maxNoImproveGen} Generationen.')
             break
 
         if sigma < sigmaImprove:
-            print(f'    [CMA-ES] Abbruch: sigma zu klein {sigma:.3g}')
+            # print(f'    [CMA-ES] Abbruch: sigma zu klein {sigma:.3g}')
             break
 
-
-
         # Cumulation: Update evolution paths
         ps = (1 - cs) * ps + np.sqrt(cs * (2 - cs) * mueff) * (B @ zmean)
         norm_ps = np.linalg.norm(ps)
-        hsig = norm_ps / np.sqrt(1 - (1 - cs)**(2 * counteval / lambda_)) / chiN < \
-               (1.4 + 2 / (N + 1))
+        hsig = norm_ps / np.sqrt(1 - (1 - cs) ** (2 * counteval / lambda_)) / chiN < (1.4 + 2 / (N + 1))
         hsig = 1.0 if hsig else 0.0
 
         pc = (1 - cc) * pc + hsig * np.sqrt(cc * (2 - cc) * mueff) * (B @ D @ zmean)
@@ -140,16 +137,17 @@ def escma(func, *, N=10, xmean=None, sigma=0.5, stopfitness=1e-10, stopeval=None
         # Escape flat fitness, or better terminate?
         if arfitness[0] == arfitness[int(np.ceil(0.7 * lambda_)) - 1]:
             sigma = sigma * np.exp(0.2 + cs / damps)
-            print('    [CMA-ES] stopfitness erreicht.')
-            #print("warning: flat fitness, consider reformulating the objective")
+            # print('    [CMA-ES] stopfitness erreicht.')
+            # print("warning: flat fitness, consider reformulating the objective")
+            break
 
-        #print(f"{counteval}: {arfitness[0]}")
+        # print(f"{counteval}: {arfitness[0]}")
 
-    #Final Message
-    #print(f"{counteval}: {arfitness[0]}")
+    # Final Message
+    # print(f"{counteval}: {arfitness[0]}")
     xmin = arx[:, arindex[0]]
     bestValue = arfitness[0]
-    print(f'    [CMA-ES] Ende: Gen = {gen}, best = {round(bestValue, 6)}')
+    # print(f'    [CMA-ES] Ende: Gen = {gen}, best = {round(bestValue, 6)}')
     return xmin
 
 

ES/gha1_ES.py

@@ -0,0 +1,247 @@
+from __future__ import annotations
+
+from typing import List, Tuple
+
+import numpy as np
+from numpy.typing import NDArray
+
+import winkelumrechnungen as wu
+from ES.Hansen_ES_CMA import escma
+from GHA_triaxial.gha1_ana import gha1_ana
+from GHA_triaxial.gha1_approx import gha1_approx
+from GHA_triaxial.utils import jacobi_konstante
+from ellipsoid_triaxial import EllipsoidTriaxial
+from utils_angle import wrap_mpi_pi
+
+
+def ENU_beta_omega(beta: float, omega: float, ell: EllipsoidTriaxial) \
+        -> Tuple[NDArray, NDArray, NDArray, float, float, NDArray]:
+    """
+    Analytische ENU-Basis in ellipsoidische Koordinaten (β, ω) nach Karney (2025), S. 2
+    :param beta: Beta Koordinate
+    :param omega: Omega Koordinate
+    :param ell: Ellipsoid
+    :return:    E_hat = Einheitsrichtung entlang wachsendem ω (East)
+                N_hat = Einheitsrichtung entlang wachsendem β (North)
+                U_hat = Einheitsnormale (Up)
+                En & Nn = Längen der unnormierten Ableitungen
+                R (XYZ) = Punkt in XYZ
+    """
+    # Berechnungshilfen
+    omega = wrap_mpi_pi(omega)
+    cb = np.cos(beta)
+    sb = np.sin(beta)
+    co = np.cos(omega)
+    so = np.sin(omega)
+    # D = sqrt(a^2 - c^2)
+    D = np.sqrt(ell.ax*ell.ax - ell.b*ell.b)
+    # Sx = sqrt(a^2 - b^2 sin^2β - c^2 cos^2β)
+    Sx = np.sqrt(ell.ax*ell.ax - ell.ay*ell.ay*(sb*sb) - ell.b*ell.b*(cb*cb))
+    # Sz = sqrt(a^2 sin^2ω + b^2 cos^2ω - c^2)
+    Sz = np.sqrt(ell.ax*ell.ax*(so*so) + ell.ay*ell.ay*(co*co) - ell.b*ell.b)
+
+    # Karney Gl. (4)
+    X = ell.ax * co * Sx / D
+    Y = ell.ay * cb * so
+    Z = ell.b * sb * Sz / D
+    R = np.array([X, Y, Z], dtype=float)
+
+    # --- Ableitungen - Karney Gl. (5a,b,c)---
+    # E = ∂R/∂ω
+    dX_dw = -ell.ax * so * Sx / D
+    dY_dw = ell.ay * cb * co
+    dZ_dw = ell.b * sb * (so * co * (ell.ax*ell.ax - ell.ay*ell.ay) / Sz) / D
+    E = np.array([dX_dw, dY_dw, dZ_dw], dtype=float)
+
+    # N = ∂R/∂β
+    dX_db = ell.ax * co * (sb * cb * (ell.b*ell.b - ell.ay*ell.ay) / Sx) / D
+    dY_db = -ell.ay * sb * so
+    dZ_db = ell.b * cb * Sz / D
+    N = np.array([dX_db, dY_db, dZ_db], dtype=float)
+
+    # U = Grad(x^2/a^2 + y^2/b^2 + z^2/c^2 - 1)
+    U = np.array([X/(ell.ax*ell.ax), Y/(ell.ay*ell.ay), Z/(ell.b*ell.b)], dtype=float)
+
+    En = float(np.linalg.norm(E))
+    Nn = float(np.linalg.norm(N))
+    Un = float(np.linalg.norm(U))
+
+    N_hat = N / Nn
+    E_hat = E / En
+    U_hat = U / Un
+    E_hat -= float(np.dot(E_hat, N_hat)) * N_hat
+    E_hat = E_hat / max(np.linalg.norm(E_hat), 1e-18)
+
+    return E_hat, N_hat, U_hat, En, Nn, R
+
+
+def azimuth_at_ESpoint(P_prev: NDArray, P_curr: NDArray, E_hat_curr: NDArray, N_hat_curr: NDArray, U_hat_curr: NDArray) -> float:
+    """
+    Berechnet das Azimut in der lokalen Tangentialebene am aktuellen Punkt P_curr, gemessen
+    an der Bewegungsrichtung vom vorherigen Punkt P_prev nach P_curr.
+    :param P_prev: vorheriger Punkt
+    :param P_curr: aktueller Punkt
+    :param E_hat_curr: Einheitsvektor der lokalen Tangentialrichtung am Punkt P_curr
+    :param N_hat_curr: Einheitsvektor der lokalen Tangentialrichtung am Punkt P_curr
+    :param U_hat_curr: Einheitsnormalenvektor am Punkt P_curr
+    :return: Azimut in Radiant
+    """
+    v = (P_curr - P_prev).astype(float)
+    vT = v - float(np.dot(v, U_hat_curr)) * U_hat_curr
+    vTn = max(np.linalg.norm(vT), 1e-18)
+    vT_hat = vT / vTn
+    sE = float(np.dot(vT_hat, E_hat_curr))
+    sN = float(np.dot(vT_hat, N_hat_curr))
+
+    return wrap_mpi_pi(float(np.arctan2(sE, sN)))
+
+
+def optimize_next_point(beta_i: float, omega_i: float, alpha_i: float, ds: float, gamma0: float,
+                        ell: EllipsoidTriaxial, maxSegLen: float = 1000.0, sigma0: float = None) -> Tuple[float, float, NDArray, float]:
+    """
+    Berechnung der 1. GHA mithilfe der CMA-ES.
+    Die CMA-ES optimiert sukzessive einen Punkt, der maxSegLen vom vorherigen Punkt entfernt und zusätzlich auf der
+    geodätischen Linien liegt. Somit entsteht ein Geodäten ähnlicher Polygonzug auf der Oberfläche des dreiachsigen Ellipsoids.
+    :param beta_i: Beta Koordinate am Punkt i
+    :param omega_i: Omega Koordinate am Punkt i
+    :param alpha_i: Azimut am Punkt i
+    :param ds: Gesamtlänge
+    :param gamma0: Jacobi-Konstante am Startpunkt
+    :param ell: Ellipsoid
+    :param maxSegLen: maximale Segmentlänge
+    :param sigma0:
+    :return:
+    """
+    # Startbasis
+    E_i, N_i, U_i, En_i, Nn_i, P_i = ENU_beta_omega(beta_i, omega_i, ell)
+    # Prediktor: dβ ≈ ds cosα / |N|, dω ≈ ds sinα / |E|
+
+    En_eff = max(En_i, 1e-9)
+    Nn_eff = max(Nn_i, 1e-9)
+
+    d_beta = ds * np.cos(alpha_i) / Nn_eff
+    d_omega = ds * np.sin(alpha_i) / En_eff
+
+    # optional: harte Schritt-Clamps (verhindert wrap-chaos)
+    d_beta = float(np.clip(d_beta, -0.2, 0.2))  # rad
+    d_omega = float(np.clip(d_omega, -0.2, 0.2))  # rad
+
+    # d_beta = ds * float(np.cos(alpha_i)) / Nn_i
+    # d_omega = ds * float(np.sin(alpha_i)) / En_i
+    beta_pred = beta_i + d_beta
+    omega_pred = wrap_mpi_pi(omega_i + d_omega)
+
+    xmean = np.array([beta_pred, omega_pred], dtype=float)
+
+    if sigma0 is None:
+        R0 = (ell.ax + ell.ay + ell.b) / 3
+        sigma0 = 1e-5 * (ds / R0)
+
+    def fitness(x: NDArray) -> float:
+        """
+        Fitnessfunktion: Fitnesscheck erfolgt anhand der Segmentlänge und der Jacobi-Konstante.
+        Die Segmentlänge muss möglichst gut zum Sollwert passen. Die Jacobi-Konstante am Punkt x muss zur
+        Jacobi-Konstanten am Startpunkt passen, damit der Polygonzug auf derselben geodätischen Linie bleibt.
+        :param x: Koordinate in beta, lambda aus der CMA-ES
+        :return: Fitnesswert (f)
+        """
+        beta = x[0]
+        omega = wrap_mpi_pi(x[1])
+
+        P = ell.ell2cart_karney(beta, omega)  # in kartesischer Koordinaten
+        d = float(np.linalg.norm(P - P_i))  # Distanz zwischen
+
+        # maxSegLen einhalten
+        J_len = ((d - ds) / ds) ** 2
+        w_len = 1.0
+
+        # Azimut für Jacobi-Konstante
+        E_j, N_j, U_j, _, _, _ = ENU_beta_omega(beta, omega, ell)
+        alpha_end = azimuth_at_ESpoint(P_i, P, E_j, N_j, U_j)
+
+        # Jacobi-Konstante
+        g_end = jacobi_konstante(beta, omega, alpha_end, ell)
+        J_gamma = (g_end - gamma0) ** 2
+        w_gamma = 10
+
+        f = float(w_len * J_len + w_gamma * J_gamma)
+
+        return f
+
+    xb = escma(fitness, N=2, xmean=xmean, sigma=sigma0)  # Aufruf CMA-ES
+
+    beta_best = xb[0]
+    omega_best = wrap_mpi_pi(xb[1])
+    P_best = ell.ell2cart_karney(beta_best, omega_best)
+    E_j, N_j, U_j, _, _, _ = ENU_beta_omega(beta_best, omega_best, ell)
+    alpha_end = azimuth_at_ESpoint(P_i, P_best, E_j, N_j, U_j)
+
+    return beta_best, omega_best, P_best, alpha_end
+
+
+def gha1_ES(ell: EllipsoidTriaxial, beta0: float, omega0: float, alpha0: float, s_total: float, maxSegLen: float = 1000, all_points: bool = False)\
+        -> Tuple[NDArray, float, NDArray] | Tuple[NDArray, float]:
+    """
+    Aufruf der 1. GHA mittels CMA-ES
+    :param ell: Ellipsoid
+    :param beta0: Beta Startkoordinate
+    :param omega0: Omega Startkoordinate
+    :param alpha0: Azimut Startkoordinate
+    :param s_total: Gesamtstrecke
+    :param maxSegLen: maximale Segmentlänge
+    :param all_points: Alle Punkte ausgeben?
+    :return: Zielpunkt Pk, Azimut am Zielpunkt und Punktliste
+    """
+    beta = float(beta0)
+    omega = wrap_mpi_pi(float(omega0))
+    alpha = wrap_mpi_pi(float(alpha0))
+
+    gamma0 = jacobi_konstante(beta, omega, alpha, ell)  # Referenz-γ0
+
+    P_all: List[NDArray] = [ell.ell2cart_karney(beta, omega)]
+    alpha_end: List[float] = [alpha]
+
+    s_acc = 0.0
+    step = 0
+    nsteps_est = int(np.ceil(s_total / maxSegLen))
+    while s_acc < s_total - 1e-9:
+        step += 1
+        ds = min(maxSegLen, s_total - s_acc)
+        # print(f"[GHA1-ES] Step {step}/{nsteps_est}  ds={ds:.3f} m  s_acc={s_acc:.3f} m  beta={beta:.6f}  omega={omega:.6f}  alpha={alpha:.6f}")
+
+        beta, omega, P, alpha = optimize_next_point(beta_i=beta, omega_i=omega, alpha_i=alpha, ds=ds, gamma0=gamma0,
+                                                    ell=ell, maxSegLen=maxSegLen)
+        s_acc += ds
+        P_all.append(P)
+        alpha_end.append(wrap_mpi_pi(alpha))
+        if step > nsteps_est + 50:
+            raise RuntimeError("GHA1_ES: Zu viele Schritte – vermutlich Konvergenzproblem / falsche Azimut-Konvention.")
+    Pk = P_all[-1]
+    alpha1 = float(alpha_end[-1])
+
+    if all_points:
+        return Pk, alpha1, np.array(P_all)
+    else:
+        return Pk, alpha1
+
+
+if __name__ == "__main__":
+    ell = EllipsoidTriaxial.init_name("BursaSima1980round")
+    s = 180000
+    # alpha0 = 3
+    alpha0 = wu.gms2rad([5, 0, 0])
+    beta = 0
+    omega = 0
+    P0 = ell.ell2cart(beta, omega)
+    point1, alpha1 = gha1_ana(ell, P0, alpha0=alpha0, s=s, maxM=100, maxPartCircum=32)
+    point1app, alpha1app = gha1_approx(ell, P0, alpha0=alpha0, s=s, ds=1000)
+
+    res, alpha, points = gha1_ES(ell, beta0=beta, omega0=-omega, alpha0=alpha0, s_total=s, maxSegLen=1000)
+
+    print(point1)
+    print(res)
+    print(alpha)
+    print(points)
+    # print("alpha1 (am Endpunkt):", res.alpha1)
+    print(res - point1)
+    print(point1app - point1, "approx")

ES/gha2_ES.py

@@ -0,0 +1,182 @@
+from typing import Tuple
+
+import numpy as np
+import plotly.graph_objects as go
+from numpy.typing import NDArray
+
+from ES.Hansen_ES_CMA import escma
+from GHA_triaxial.gha2_num import gha2_num
+from GHA_triaxial.utils import sigma2alpha
+from ellipsoid_triaxial import EllipsoidTriaxial
+
+
+def Sehne(P1: NDArray, P2: NDArray) -> float:
+    """
+    Berechnung der 3D-Distanz zwischen zwei kartesischen Punkten
+    :param P1: kartesische Koordinate Punkt 1
+    :param P2: kartesische Koordinate Punkt 2
+    :return: Bogenlänge s
+    """
+    R12 = P2-P1
+    s = float(np.linalg.norm(R12))
+
+    return s
+
+
+def gha2_ES(ell: EllipsoidTriaxial, P0: NDArray, Pk: NDArray, maxSegLen: float = None, all_points: bool = False) -> Tuple[float, float, float, NDArray] | Tuple[float, float, float]:
+    """
+    Berechnen der 2. GHA mithilfe der CMA-ES.
+    Die CMA-ES optimiert sukzessive den Mittelpunkt zwischen Start- und Zielpunkt. Der Abbruch der Berechnung erfolgt, wenn alle Segmentlängen <= maxSegLen sind.
+    Die Distanzen zwischen den einzelnen Punkten werden als direkte 3D-Distanzen berechnet und aufaddiert.
+    :param ell: Ellipsoid
+    :param P0: Startpunkt
+    :param Pk: Zielpunkt
+    :param maxSegLen: maximale Segmentlänge
+    :param all_points: Ergebnisliste mit allen Punkte, die wahlweise mit ausgegeben wird
+    :return: Richtungswinkel in RAD des Start- und Zielpunktes und Gesamtlänge
+    """
+    P_left: NDArray = None
+    P_right: NDArray = None
+
+    def midpoint_fitness(x: tuple) -> float:
+        """
+        Fitness für einen Mittelpunkt P_middle zwischen P_left und P_right auf dem triaxialen Ellipsoid:
+        - Minimiert d(P_left, P_middle) + d(P_middle, P_right)
+        - Erzwingt d(P_left,P_middle) ≈ d(P_middle,P_right)  (echter Mittelpunkt im Sinne der Polygonkette)
+        :param x: enthält die Startwerte von u und v
+        :return: Fitnesswert (f)
+        """
+        nonlocal P_left, P_right, ell
+
+        u, v = x
+        P_middle = ell.para2cart(u, v)
+        d1 = Sehne(P_left, P_middle)
+        d2 = Sehne(P_middle, P_right)
+        base = d1 + d2
+
+        # midpoint penalty (dimensionslos)
+        # relative Differenz, skaliert über verschiedene Segmentlängen
+        denom = max(base, 1e-9)
+        pen_equal = ((d1 - d2) / denom) ** 2
+        w_equal = 10.0
+
+        f = base + denom * w_equal * pen_equal
+
+        return f
+
+    R0 = (ell.ax + ell.ay + ell.b) / 3
+    if maxSegLen is None:
+        maxSegLen = R0 * 1 / (637.4*2)  # 10km Segment bei mittleren Erdradius
+
+    sigma_uv_nom = 1e-3 * (maxSegLen / R0)  # ~1e-5
+    points: list[NDArray] = [P0, Pk]
+    startIter = 0
+    level = 0
+
+    while True:
+        seg_lens = [Sehne(points[i], points[i+1]) for i in range(len(points)-1)]
+        max_len = max(seg_lens)
+        if max_len <= maxSegLen:
+            break
+
+        level += 1
+        new_points: list[NDArray] = [points[0]]
+        for i in range(len(points) - 1):
+            A = points[i]
+            B = points[i+1]
+            dAB = Sehne(A, B)
+            # print(dAB)
+
+            if dAB > maxSegLen:
+                # global P_left, P_right
+                P_left, P_right = A, B
+                Au, Av = ell.cart2para(A)
+                Bu, Bv = ell.cart2para(B)
+                u0 = (Au + Bu) / 2
+                v0 = Av + 0.5 * np.arctan2(np.sin(Bv - Av), np.cos(Bv - Av))
+                xmean = [u0, v0]
+
+                sigmaStep = sigma_uv_nom * (Sehne(A, B) / maxSegLen)
+
+                u, v = escma(midpoint_fitness, N=2, xmean=xmean, sigma=sigmaStep)  # Aufruf CMA-ES
+
+                P_next = ell.para2cart(u, v)
+                new_points.append(P_next)
+                startIter += 1
+                maxIter = 10000
+                if startIter > maxIter:
+                    raise RuntimeError("GHA2_ES: maximale Iterationen überschritten")
+            new_points.append(B)
+
+        points = new_points
+        # print(f"[Level {level}] Punkte: {len(points)} | max Segment: {max_len:.3f} m")
+
+    P_all = np.vstack(points)
+    totalLen = float(np.sum(np.linalg.norm(P_all[1:] - P_all[:-1], axis=1)))
+
+    if len(points) >= 3:
+        p0i = ell.point_onto_ellipsoid(P0 + 10.0 * (points[1] - P0) / np.linalg.norm(points[1] - P0))
+        sigma0 = (p0i - P0) / np.linalg.norm(p0i - P0)
+        alpha0 = sigma2alpha(ell, sigma0, P0)
+
+        p1i = ell.point_onto_ellipsoid(Pk - 10.0 * (Pk - points[-2]) / np.linalg.norm(Pk - points[-2]))
+        sigma1 = (Pk - p1i) / np.linalg.norm(Pk - p1i)
+        alpha1 = sigma2alpha(ell, sigma1, Pk)
+    else:
+        alpha0 = None
+        alpha1 = None
+
+    if all_points:
+        return alpha0, alpha1, totalLen, P_all
+    return alpha0, alpha1, totalLen
+
+
+def show_points(points: NDArray, pointsES: NDArray, p0: NDArray, p1: NDArray):
+    """
+    Anzeigen der Punkte
+    :param points: wahre Punkte der Linie
+    :param pointsES: Punkte der Linie aus ES
+    :param p0: wahrer Startpunkt
+    :param p1: wahrer Endpunkt
+    """
+    fig = go.Figure()
+
+    fig.add_scatter3d(x=pointsES[:, 0], y=pointsES[:, 1], z=pointsES[:, 2],
+                      mode='lines', line=dict(color="green", width=3), name="Numerisch")
+    fig.add_scatter3d(x=points[:, 0], y=points[:, 1], z=points[:, 2],
+                      mode='lines', line=dict(color="red", width=3), name="ES")
+    fig.add_scatter3d(x=[p0[0]], y=[p0[1]], z=[p0[2]],
+                      mode='markers', marker=dict(color="black"), name="P0")
+    fig.add_scatter3d(x=[p1[0]], y=[p1[1]], z=[p1[2]],
+                      mode='markers', marker=dict(color="black"), name="P1")
+
+    fig.update_layout(
+        scene=dict(xaxis_title='X [km]',
+                   yaxis_title='Y [km]',
+                   zaxis_title='Z [km]',
+                   aspectmode='data'))
+
+    fig.show()
+
+
+if __name__ == '__main__':
+    ell = EllipsoidTriaxial.init_name("Bursa1970")
+
+    beta0, lamb0 = (0.2, 0.1)
+    P0 = ell.ell2cart(beta0, lamb0)
+    beta1, lamb1 = (0.3, 0.2)
+    P1 = ell.ell2cart(beta1, lamb1)
+
+    alpha0, alpha1, s_num, betas, lambs = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=1000, all_points=True)
+    points_num = []
+    for beta, lamb in zip(betas, lambs):
+        points_num.append(ell.ell2cart(beta, lamb))
+    points_num = np.array(points_num)
+
+    alpha0, alpha1, s, points = gha2_ES(ell, P0, P1)
+    print(s_num)
+    print(s)
+    print(alpha0)
+    print(alpha1)
+    print(s - s_num)
+    show_points(points, points_num, P0, P1)

GHA_biaxial/bessel.py

@@ -1,25 +0,0 @@
-from numpy import *
-import scipy as sp
-
-
-def gha1(re, phi_p1, lambda_p1, A_p1, s):
-    psi_p1 = re.phi2psi(phi_p1)
-    A_0 = arcsin(cos(psi_p1) * sin(A_p1))
-    temp = sin(psi_p1) / cos(A_0)
-    sigma_p1 = arcsin(sin(psi_p1) / cos(A_0))
-
-    sqrt_sigma = lambda sigma: sqrt(1 + re.e_ ** 2 * cos(A_0) ** 2 * sin(sigma) ** 2)
-    int_sqrt_sigma = lambda sigma: sp.integrate.quad(sqrt_sigma, sigma_p1, sigma)[0]
-
-    f_sigma_p2i = lambda sigma_p2i: (int_sqrt_sigma(sigma_p2i) - s / re.b)
-
-    sigma_p2_0 = sigma_p1 + s / re.a
-    sigma_p2 = sp.optimize.newton(f_sigma_p2i, sigma_p2_0)
-    psi_p2 = arcsin(cos(A_0) * sin(sigma_p2))
-    phi_p2 = re.psi2phi(psi_p2)
-    A_p2 = arcsin(sin(A_0) / cos(psi_p2))
-
-    f_d_lambda = lambda sigma: sin(A_0) * sqrt_sigma(sigma) / (1 - cos(A_0)**2 * sin(sigma)**2)
-    d_lambda = sqrt(1-re.e**2) * sp.integrate.quad(f_d_lambda, sigma_p1, sigma_p2)[0]
-    lambda_p2 = lambda_p1 + d_lambda
-    return phi_p2, lambda_p2, A_p2

GHA_biaxial/gauss.py

@@ -1,107 +0,0 @@
-from numpy import sin, cos, pi, sqrt, tan, arcsin, arccos, arctan
-import ausgaben as aus
-
-
-def gha1(re, phi_p1, lambda_p1, A_p1, s, eps):
-    """
-    Berechnung der 1. Geodätische Hauptaufgabe nach Gauß´schen Mittelbreitenformeln
-    :param re: Klasse Ellipsoid
-    :param phi_p1: Breite Punkt 1
-    :param lambda_p1: Länge Punkt 1
-    :param A_p1: Azimut der geodätischen Linie in Punkt 1
-    :param s: Strecke zu Punkt 2
-    :param eps: Abbruchkriterium für Winkelgrößen
-    :return: Breite, Länge, Azimut von Punkt 2
-    """
-
-    t = lambda phi: tan(phi)
-    eta = lambda phi: sqrt(re.e_ ** 2 * cos(phi) ** 2)
-
-    F1 = lambda A, phi, s: 1 + (2 + 3 * t(phi)**2 + 2 * eta(phi)**2) / (24 * re.N(phi) ** 2) * sin(A) ** 2 * s ** 2 \
-                           + (t(phi)**2 - 1) * eta(phi) ** 2 / (8 * re.N(phi) ** 2) * cos(A) ** 2 * s ** 2
-
-    F2 = lambda A, phi, s: 1 + t(phi) ** 2 / (24 * re.N(phi) ** 2) * sin(A) ** 2 * s ** 2 \
-                           - (1 + eta(phi)**2 - 9 * eta(phi)**2 * t(phi)**2) / (24 * re.N(phi) ** 2) * cos(A) ** 2 * s ** 2
-
-    F3 = lambda A, phi, s: 1 + (1 + eta(phi)**2) / (12 * re.N(phi) ** 2) * sin(A) ** 2 * s ** 2 \
-                           + (3 + 8 * eta(phi)**2) / (24 * re.N(phi) ** 2) * cos(A) ** 2 * s ** 2
-
-    phi_p2_i = lambda A, phi: phi_p1 + cos(A) / re.M(phi) * s * F1(A, phi, s)
-    lambda_p2_i = lambda A, phi: lambda_p1 + sin(A) / (re.N(phi) * cos(phi)) * s * F2(A, phi, s)
-    A_p2_i = lambda A, phi: A_p1 + sin(A) * tan(phi) / re.N(phi) * s * F3(A, phi, s)
-
-    phi_p0_i = lambda phi2: (phi_p1 + phi2) / 2
-    A_p1_i = lambda A2: (A_p1 + A2) / 2
-
-    phi_p0 = []
-    A_p0 = []
-    phi_p2 = []
-    lambda_p2 = []
-    A_p2 = []
-
-    # 1. Näherung für P2
-    phi_p2.append(phi_p1 + cos(A_p1) / re.M(phi_p1) * s)
-    lambda_p2.append(lambda_p1 + sin(A_p1) / (re.N(phi_p1) * cos(phi_p1)) * s)
-    A_p2.append(A_p1 + sin(A_p1) * tan(phi_p1) / re.N(phi_p1) * s)
-
-    while True:
-        # Berechnug P0 durch Mittelbildung
-        phi_p0.append(phi_p0_i(phi_p2[-1]))
-        A_p0.append(A_p1_i(A_p2[-1]))
-        # Berechnung P2
-        phi_p2.append(phi_p2_i(A_p0[-1], phi_p0[-1]))
-        lambda_p2.append(lambda_p2_i(A_p0[-1], phi_p0[-1]))
-        A_p2.append(A_p2_i(A_p0[-1], phi_p0[-1]))
-
-        # Abbruchkriterium
-        if abs(phi_p2[-2] - phi_p2[-1]) < eps and \
-           abs(lambda_p2[-2] - lambda_p2[-1]) < eps and \
-           abs(A_p2[-2] - A_p2[-1]) < eps:
-            break
-
-    nks = 5
-    for i in range(len(phi_p2)):
-        print(f"P2[{i}]: {aus.gms('phi', phi_p2[i], nks)}\t{aus.gms('lambda', lambda_p2[i], nks)}\t{aus.gms('A', A_p2[i], nks)}")
-        if i != len(phi_p2)-1:
-            print(f"P0[{i}]: {aus.gms('phi', phi_p0[i], nks)}\t\t\t\t\t\t\t\t{aus.gms('A', A_p0[i], nks)}")
-    return phi_p2, lambda_p2, A_p2
-
-
-def gha2(re, phi_p1, lambda_p1, phi_p2, lambda_p2):
-    """
-    Berechnung der 2. Geodätische Hauptaufgabe nach Gauß´schen Mittelbreitenformeln
-    :param re: Klasse Ellipsoid
-    :param phi_p1: Breite Punkt 1
-    :param lambda_p1: Länge Punkt 1
-    :param phi_p2: Breite Punkt 2
-    :param lambda_p2: Länge Punkt 2
-    :return: Länge der geodätischen Linie, Azimut von P1 nach P2, Azimut von P2 nach P1
-    """
-
-    t = lambda phi: tan(phi)
-    eta = lambda phi: sqrt(re.e_ ** 2 * cos(phi) ** 2)
-
-    phi_0 = (phi_p1 + phi_p2) / 2
-    d_phi = phi_p2 - phi_p1
-    d_lambda = lambda_p2 - lambda_p1
-
-    f_A = lambda phi: (2 + 3*t(phi)**2 + 2*eta(phi)**2) / 24
-    f_B = lambda phi: ((t(phi)**2 - 1) * eta(phi)**2) / 8
-    # f_C = lambda phi: (t(phi)**2) / 24
-    f_D = lambda phi: (1 + eta(phi)**2 - 9 * eta(phi)**2 * t(phi)**2) / 24
-
-    F1 = lambda phi: d_phi * re.M(phi) * (1 - f_A(phi) * d_lambda ** 2 * cos(phi) ** 2 -
-                                          f_B(phi) * d_phi ** 2 / re.V(phi) ** 4)
-    F2 = lambda phi: d_lambda * re.N(phi) * cos(phi) * (1 - 1 / 24 * d_lambda ** 2 * sin(phi) ** 2 +
-                                                           f_D(phi) * d_phi ** 2 / re.V(phi) ** 4)
-
-    s = sqrt(F1(phi_0) ** 2 + F2(phi_0) ** 2)
-    A_0 = arctan(F2(phi_0) / F1(phi_0))
-    d_A = d_lambda * sin(phi_0) * (1 + (1 + eta(phi_0) ** 2) / 12 * s ** 2 * sin(A_0) ** 2 / re.N(phi_0) ** 2 +
-                                      (3 + 8 * eta(phi_0) ** 2) / 24 * s ** 2 * cos(A_0) ** 2 / re.N(phi_0) ** 2)
-
-    A_p1 = A_0 - d_A / 2
-    A_p2 = A_0 + d_A / 2
-    A_p2_p1 = A_p2 + pi
-
-    return s, A_p1, A_p2_p1

GHA_biaxial/rk.py

@@ -1,21 +0,0 @@
-import runge_kutta as rk
-from numpy import sin, cos, tan
-import winkelumrechnungen as wu
-from ellipsoide import EllipsoidBiaxial
-import numpy as np
-
-def gha1(re: EllipsoidBiaxial, x0, y0, z0, A0, s, num):
-    phi0, lamb0, h0 = re.cart2ell(0.001, wu.gms2rad([0, 0, 0.001]), x0, y0, z0)
-
-    def buildODE():
-        def ODE(s, v):
-            phi, lam, A = v
-            dphi = cos(A) * re.V(phi) ** 3 / re.c
-            dlam = sin(A) * re.V(phi) / (cos(phi) * re.c)
-            dA = tan(phi) * sin(A) * re.V(phi) / re.c
-            return np.array([dphi, dlam, dA])
-        return ODE
-
-    _, funktionswerte = rk.rk4(buildODE(), 0, np.array([phi0, lamb0, A0]), s, num)
-    coords = re.ell2cart(funktionswerte[-1][0], funktionswerte[-1][1], h0)
-    return coords

GHA_triaxial/ES_gha2.py

@@ -1,228 +0,0 @@
-import numpy as np
-from numpy import arccos
-from Hansen_ES_CMA import escma
-from ellipsoide import EllipsoidTriaxial
-from numpy.typing import NDArray
-import plotly.graph_objects as go
-from GHA_triaxial.panou_2013_2GHA_num import gha2_num
-from utils import sigma2alpha
-
-ell_ES: EllipsoidTriaxial = None
-P_start: NDArray = None
-P_prev: NDArray = None
-P_next: NDArray = None
-P_end: NDArray = None
-stepLen: float = None
-
-def Bogenlaenge(P1: NDArray, P2: NDArray) -> float:
-    """
-    Berechnung der mittleren Bogenlänge zwischen zwei kartesischen Punkten
-    :param P1: kartesische Koordinate Punkt 1
-    :param P2: kartesische Koordinate Punkt 2
-    :return: Bogenlänge s
-    """
-    R1 = np.linalg.norm(P1)
-    R2 = np.linalg.norm(P2)
-    R = 0.5 * (R1 + R2)
-    theta = arccos(P1 @ P2 / (R1 * R2))
-    s = float(R * theta)
-    return s
-
-
-def gha2_ES(ell: EllipsoidTriaxial, P0: NDArray, Pk: NDArray, stepLenTarget: float = None, sigmaStep: float = 1e-5, stopeval: int = 1000, maxSteps: int = 10000, all_points: bool = False):
-    """
-    Berechnen der 2. GHA mithilfe der CMA-ES.
-    Die CMA-ES optimiert sukzessive einzelne Punkte, die einen definierten Abstand (stepLenTarget) zum vorherigen und den kürzesten
-    Abstand zum Zielpunkt aufweisen. Der Abbruch der Optimierung erfolgt, wenn die Restdistanz zwischen vorherigen und Zielpunkt die
-    'stepLenTarget' unterschreitet. Die Distanzen zwischen den einzelnen Punkten werden als mittlere Bogenlänge berechnet und aufaddiert.
-    :param ell: Parameter des triaxialen Ellipsoids
-    :param P0: Startpunkt
-    :param Pk: Zielpunkt
-    :param stepLenTarget: Abstand zwischen vorherigen und optimierten Punkt
-    :param sigmaStep: Sigma Startwert für die CMA-ES (Suchraum um den zu optimierenden Punkt)
-    :param stopeval: maximale Iterationen
-    :param maxSteps: maximale Anzahl der zu optimierenden Punkte
-    :param all_points: Ergebnisliste mit allen Punkte, die wahlweise mit ausgegeben werden kann
-    :return: Richtungswinkel des Start- und Zielpunktes, totalLen
-    """
-    global ell_ES, P_start, P_prev, P_next, P_end, stepLen
-    ell_ES = ell
-    P_start = P0
-    P_end = Pk
-    if stepLenTarget is None:
-        R0 = (ell.ax + ell.ay + ell.b) / 3
-        stepLenTarget = R0 * 1 / 600
-    stepLen = stepLenTarget
-
-    P_all = [P_start]
-    totalLen = 0
-
-    P_prev = P_start
-
-    for i in range(1, maxSteps):
-
-        d_remain = Bogenlaenge(P_prev, P_end)
-
-        # Abbruch: letzter "Rest-Schritt" ist < 10 km -> Ziel anhängen
-        if d_remain <= stepLen:
-            P_all.append(P_end)
-            totalLen += d_remain
-            print(f'[Punkt {i}] Stop: Restdistanz {round(d_remain, 3)} m <= {round(stepLen, 3)} m, Ziel angehängt.')
-            break
-
-        # Globals für Fitness: aktueller Start(P0) und Ziel(Pk)
-        # P0 = P_prev;
-        # Pk = P_end;
-
-        # Näherung für die ES ermitteln
-        # % d = P_end - P_prev;
-        # % L = Bogenlaenge(P_prev, P_end);
-        # % t = min(stepLen / L, 1.0);
-        # % Q = P_prev + t * d;
-        # % q = [Q(1) / a;
-        # Q(2) / b;
-        # Q(3) / c];
-        # % nq = norm(q);
-        # % q = q / nq;
-
-        # %q
-        # entspricht[cos(u)
-        # cos(v);
-        # cos(u)
-        # sin(v);
-        # sin(u)] auf
-        # Einheitskugel
-        #arg_u = max(-1, min(1, q(3)));
-        #
-        # %Quadrantenabfrage
-        #u0 = mod(asin(arg_u) + pi / 2, pi) - pi / 2;
-        #v0 = atan2(q(2), q(1));
-        #xmean_init = [u0;v0];
-        xmean_init = ell.point_onto_ellipsoid(P_prev + stepLen * (P_end - P_prev) / np.linalg.norm(P_end - P_prev))
-
-        # [~, ~, aux] = geoLength(xmean_init);
-        # print('Startguess: d_step=%.3f (soll %.3f), d_to_target=%.3f\n', aux(1), stepLen, aux(2));
-
-        print(f'[Punkt {i}] Optimiere nächsten Punkt: Restdistanz = {round(d_remain, 3)} m')
-        xmean_init = np.array(ell_ES.cart2para(xmean_init))
-
-        u, v = escma(geoLength, N=2, xmean=xmean_init, sigma=sigmaStep, stopfitness=-np.inf, stopeval=stopeval)
-
-        P_next = ell.para2cart(u, v)
-
-        d_step = Bogenlaenge(P_prev, P_next)
-        d_new = Bogenlaenge(P_next, P_end)
-        d_old = Bogenlaenge(P_prev, P_end)
-
-        print(f'[Punkt {i}] Ergebnis: Schritt = {round(d_step, 3)} m (soll {round(stepLen, 3)}), Rest neu = {round(d_new, 3)} m')
-
-        # Sicherheitscheck: wenn wir nicht näher kommen, abbrechen
-        if d_new >= d_old:
-            print(f'Punkt {i}: Neuer Punkt ist nicht naeher am Ziel ({d_old} -> {d_new}). Abbruch.')
-            P_all.append(P_end)
-            totalLen += Bogenlaenge(P_prev, P_end)
-            break
-
-        P_all.append(P_next)
-        totalLen += d_step
-        P_prev = P_next
-
-    print('Maximale Schrittanzahl erreicht.')
-    P_all.append(P_end)
-    totalLen += Bogenlaenge(P_prev, P_end)
-
-    p0i = ell.point_onto_ellipsoid(P0 + 10 * (P_all[1] - P0) / np.linalg.norm(P_all[1] - P0))
-    sigma0 = (p0i - P0) / np.linalg.norm(p0i - P0)
-    alpha0 = sigma2alpha(ell_ES, sigma0, P0)
-
-    p1i = ell.point_onto_ellipsoid(Pk - 10 * (Pk - P_all[-2]) / np.linalg.norm(Pk - P_all[-2]))
-    sigma1 = (Pk - p1i) / np.linalg.norm(Pk - p1i)
-    alpha1 = sigma2alpha(ell_ES, sigma1, Pk)
-
-    if all_points:
-        return alpha0, alpha1, totalLen, np.array(P_all)
-    else:
-        return alpha0, alpha1, totalLen
-
-
-def geoLength(P_candidate: Tuple) -> float:
-    """
-    Berechung der Fitness eines Kandidaten anhand der Strecken
-    :param P_candidate: Kandidat in parametrischen Koordinaten
-    :return: Fitness-Wert
-    """
-    # P_candidate = [u;v] des naechsten Punktes.
-    # Ziel: Distanz zum Ziel  minimieren, aber Schrittlaenge ~ stepLenTarget erzwingen.
-    u, v = P_candidate
-
-    global ell_ES, P_start, P_prev, P_next, P_end, stepLen
-    # Punkt auf Ellipsoid
-    P_next = ell_ES.para2cart(u, v)
-
-    # Distanzen
-    d_step = Bogenlaenge(P_prev, P_next)
-    d_to_target = Bogenlaenge(P_end, P_next)
-    d_prev_to_target = Bogenlaenge(P_end, P_prev)
-
-    # Penalties(dimensionslos)
-    pen_step = ((d_step - stepLen) / stepLen)**2
-
-    # falls Punkt "weg" vom Ziel geht, extra bestrafen
-    pen_away = max(0.0, (d_to_target - d_prev_to_target) / stepLen)**2
-
-    # Gewichtungen
-    alpha = 1e2
-    # Schrittlaenge sehr hart
-    gamma = 1e2  # "nicht weg vom Ziel" hart
-
-    f = d_to_target * (1 + alpha * pen_step + gamma * pen_away)
-
-    # Für Debug / Extraktion
-    # aux = [d_step, d_to_target]
-    return f  # , P_candidate, aux
-
-def show_points(points: NDArray, pointsES: NDArray, p0: NDArray, p1: NDArray):
-    """
-    Anzeigen der Punkte
-    :param points: wahre Punkte der Linie
-    :param pointsES: Punkte der Linie aus ES
-    :param p0: wahrer Startpunkt
-    :param p1: wahrer Endpunkt
-    """
-    fig = go.Figure()
-
-    fig.add_scatter3d(x=pointsES[:, 0], y=pointsES[:, 1], z=pointsES[:, 2],
-                      mode='lines', line=dict(color="green", width=3), name="Numerisch")
-    fig.add_scatter3d(x=points[:, 0], y=points[:, 1], z=points[:, 2],
-                      mode='lines', line=dict(color="red", width=3), name="ES")
-    fig.add_scatter3d(x=[p0[0]], y=[p0[1]], z=[p0[2]],
-                      mode='markers', marker=dict(color="black"), name="P0")
-    fig.add_scatter3d(x=[p1[0]], y=[p1[1]], z=[p1[2]],
-                      mode='markers', marker=dict(color="black"), name="P1")
-
-    fig.update_layout(
-        scene=dict(xaxis_title='X [km]',
-                   yaxis_title='Y [km]',
-                   zaxis_title='Z [km]',
-                   aspectmode='data'))
-
-    fig.show()
-
-
-if __name__ == '__main__':
-    ell = EllipsoidTriaxial.init_name("Fiction")
-
-    beta0, lamb0 = (0.2, 0.1)
-    P0 = ell.ell2cart(beta0, lamb0)
-    beta1, lamb1 = (0.7, 0.3)
-    P1 = ell.ell2cart(beta1, lamb1)
-
-    alpha0, alpha1, s_num, betas, lambs = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=10000, all_points=True)
-    points_num = []
-    for beta, lamb in zip(betas, lambs):
-        points_num.append(ell.ell2cart(beta, lamb))
-    points_num = np.array(points_num)
-
-    alpha0, alpha1, s, points = gha2_ES(ell, P0, P1, all_points=True, sigmaStep=1e-5)
-    print(s - s_num)
-    show_points(points, points_num, P0, P1)

GHA_triaxial/gha1_ana.py

@@ -0,0 +1,145 @@
+import math
+from math import comb
+from typing import Tuple
+
+import numpy as np
+from numpy import arctan2, cos, sin
+from numpy.typing import NDArray
+
+import winkelumrechnungen as wu
+from GHA_triaxial.utils import pq_para
+from ellipsoid_triaxial import EllipsoidTriaxial
+from utils_angle import wrap_0_2pi
+
+
+def gha1_ana_step(ell: EllipsoidTriaxial, point: NDArray, alpha0: float, s: float, maxM: int) -> Tuple[NDArray, float]:
+    """
+    Panou, Korakitits 2020, 5ff.
+    :param ell: Ellipsoid
+    :param point: Punkt in kartesischen Koordinaten
+    :param alpha0: Azimut im Startpunkt
+    :param s: Strecke
+    :param maxM: maximale Ordnung
+    :return: Zwischenpunkt, Azimut im Zwischenpunkt
+    """
+    x, y, z = point
+
+    # S. 6
+    x_m = [x]
+    y_m = [y]
+    z_m = [z]
+
+    p, q = pq_para(ell, point)
+
+    # 48-50
+    x_m.append(p[0] * sin(alpha0) + q[0] * cos(alpha0))
+    y_m.append(p[1] * sin(alpha0) + q[1] * cos(alpha0))
+    z_m.append(p[2] * sin(alpha0) + q[2] * cos(alpha0))
+
+    # 34
+    H_ = lambda p: np.sum([comb(p, p - i) * (x_m[p - i] * x_m[i] +
+                                             1 / (1 - ell.ee ** 2) ** 2 * y_m[p - i] * y_m[i] +
+                                             1 / (1 - ell.ex ** 2) ** 2 * z_m[p - i] * z_m[i])
+                           for i in range(0, p + 1)])
+
+    # 35
+    h_ = lambda q: np.sum([comb(q, q-j) * (x_m[q-j+1] * x_m[j+1] +
+                                           1 / (1 - ell.ee ** 2) * y_m[q-j+1] * y_m[j+1] +
+                                           1 / (1 - ell.ex ** 2) * z_m[q-j+1] * z_m[j+1])
+                           for j in range(0, q+1)])
+
+    # 31
+    hH_ = lambda t: 1/H_(0) * (h_(t) - np.sum([comb(t, l-1) * H_(t+1-l) * hH_t[l-1] for l in range(1, t+1)]))
+
+    # 28-30
+    x_ = lambda m: - np.sum([comb(m-2, k) * hH_t[m-2-k] * x_m[k] for k in range(0, m-2+1)])
+    y_ = lambda m: -1 / (1-ell.ee**2) * np.sum([comb(m-2, k) * hH_t[m-2-k] * y_m[k] for k in range(0, m-2+1)])
+    z_ = lambda m: -1 / (1-ell.ex**2) * np.sum([comb(m-2, k) * hH_t[m-2-k] * z_m[k] for k in range(0, m-2+1)])
+
+    hH_t = []
+    a_m = []
+    b_m = []
+    c_m = []
+    for m in range(0, maxM+1):
+        if m >= 2:
+            hH_t.append(hH_(m-2))
+            x_m.append(x_(m))
+            y_m.append(y_(m))
+            z_m.append(z_(m))
+        fact_m = math.factorial(m)
+
+        # 22-24
+        a_m.append(x_m[m] / fact_m)
+        b_m.append(y_m[m] / fact_m)
+        c_m.append(z_m[m] / fact_m)
+
+    # 19-21
+    x_s = 0
+    for a in reversed(a_m):
+        x_s = x_s * s + a
+    y_s = 0
+    for b in reversed(b_m):
+        y_s = y_s * s + b
+    z_s = 0
+    for c in reversed(c_m):
+        z_s = z_s * s + c
+
+    p1 = np.array([x_s, y_s, z_s])
+    p_s, q_s = pq_para(ell, p1)
+
+    # 57-59
+    dx_s = 0
+    for i, a in reversed(list(enumerate(a_m[1:], start=1))):
+        dx_s = dx_s * s + i * a
+
+    dy_s = 0
+    for i, b in reversed(list(enumerate(b_m[1:], start=1))):
+        dy_s = dy_s * s + i * b
+
+    dz_s = 0
+    for i, c in reversed(list(enumerate(c_m[1:], start=1))):
+        dz_s = dz_s * s + i * c
+
+    # 52-53
+    sigma = np.array([dx_s, dy_s, dz_s])
+    P = float(p_s @ sigma)
+    Q = float(q_s @ sigma)
+
+    # 51
+    alpha1 = arctan2(P, Q)
+
+    if alpha1 < 0:
+        alpha1 += 2 * np.pi
+
+    return p1, wrap_0_2pi(alpha1)
+
+
+def gha1_ana(ell: EllipsoidTriaxial, point: NDArray, alpha0: float, s: float, maxM: int, maxPartCircum: int = 4) -> Tuple[NDArray, float]:
+    """
+    :param ell: Ellipsoid
+    :param point: Punkt in kartesischen Koordinaten
+    :param alpha0: Azimut im Startpunkt
+    :param s: Strecke
+    :param maxM: maximale Ordnung
+    :param maxPartCircum: maximale Aufteilung (1/x halber Ellipsoidumfang)
+    :return: Zielpunkt, Azimut im Zielpunkt
+    """
+    if s > np.pi / maxPartCircum * ell.ax:
+        s /= 2
+        point_step, alpha_step = gha1_ana(ell, point, alpha0, s, maxM, maxPartCircum)
+        point_end, alpha_end = gha1_ana(ell, point_step, alpha_step, s, maxM, maxPartCircum)
+    else:
+        point_end, alpha_end = gha1_ana_step(ell, point, alpha0, s, maxM)
+
+    _, _, h = ell.cart2geod(point_end, "ligas3")
+    if h > 1e-5:
+        raise Exception("GHA1_ana: explodiert, Punkt liegt nicht mehr auf dem Ellipsoid")
+
+    return point_end, wrap_0_2pi(alpha_end)
+
+
+if __name__ == "__main__":
+    ell = EllipsoidTriaxial.init_name("BursaSima1980round")
+    p0 = ell.ell2cart(wu.deg2rad(10), wu.deg2rad(20))
+    p1, alpha1 = gha1_ana(ell, p0, wu.deg2rad(36), 200000, 70)
+    print(p1, wu.rad2gms(alpha1))

GHA_triaxial/gha1_approx.py

@@ -1,10 +1,18 @@
-import numpy as np
-from ellipsoide import EllipsoidTriaxial
-from panou import louville_constant, func_sigma_ell, gha1_ana
-import plotly.graph_objects as go
-import winkelumrechnungen as wu
+from typing import Tuple
 
-def gha1_approx(ell: EllipsoidTriaxial, p0: np.ndarray, alpha0: float, s: float, ds: float, all_points: bool = False) -> Tuple[NDArray, float] | Tuple[NDArray, float, NDArray]:
+import numpy as np
+import plotly.graph_objects as go
+from numpy import cos, sin
+from numpy.typing import NDArray
+
+import winkelumrechnungen as wu
+from GHA_triaxial.utils import louville_constant, pq_ell
+from ellipsoid_triaxial import EllipsoidTriaxial
+from utils_angle import wrap_0_2pi
+
+
+def gha1_approx(ell: EllipsoidTriaxial, p0: np.ndarray, alpha0: float, s: float, ds: float, all_points: bool = False) \
+        -> Tuple[NDArray, float] | Tuple[NDArray, float, NDArray, NDArray]:
     """
     Berechung einer Näherungslösung der ersten Hauptaufgabe
     :param ell: Ellipsoid
@@ -19,27 +27,47 @@ def gha1_approx(ell: EllipsoidTriaxial, p0: np.ndarray, alpha0: float, s: float,
     points = [p0]
     alphas = [alpha0]
     s_curr = 0.0
+    last_sigma = None
+    last_p = None
+
     while s_curr < s:
         ds_step = min(ds, s - s_curr)
         if ds_step < 1e-8:
             break
+
         p1 = points[-1]
         alpha1 = alphas[-1]
-        sigma = func_sigma_ell(ell, p1, alpha1)
+
+        p, q = pq_ell(ell, p1)
+        if last_p is not None and np.dot(p, last_p) < 0:
+            p = -p
+            q = -q
+        last_p = p
+        sigma = p * sin(alpha1) + q * cos(alpha1)
+        if last_sigma is not None and np.dot(sigma, last_sigma) < 0:
+            sigma = -sigma
+            alpha1 += np.pi
+            alpha1 = wrap_0_2pi(alpha1)
         p2 = p1 + ds_step * sigma
         p2 = ell.point_onto_ellipsoid(p2)
-        ds_step = np.linalg.norm(p2 - p1)
 
-        points.append(p2)
-        dalpha = 1e-6
+        dalpha = 1e-9
         l2 = louville_constant(ell, p2, alpha1)
         dl_dalpha = (louville_constant(ell, p2, alpha1+dalpha) - l2) / dalpha
-        alpha2 = alpha1 + (l0 - l2) / dl_dalpha
-        alphas.append(alpha2)
+        if abs(dl_dalpha) < 1e-20:
+            alpha2 = alpha1 + 0
+        else:
+            alpha2 = alpha1 + (l0 - l2) / dl_dalpha
+        points.append(p2)
+        alphas.append(wrap_0_2pi(alpha2))
+
+        ds_step = np.linalg.norm(p2 - p1)
         s_curr += ds_step
+        last_sigma = sigma
+        pass
 
     if all_points:
-        return points[-1], alphas[-1], np.array(points)
+        return points[-1], alphas[-1], np.array(points), np.array(alphas)
     else:
         return points[-1], alphas[-1]
 
@@ -70,11 +98,11 @@ def show_points(points: NDArray, p0: NDArray, p1: NDArray):
 
 
 if __name__ == '__main__':
-    ell = EllipsoidTriaxial.init_name("BursaSima1980round")
-    P0 = ell.para2cart(0, 0)
-    alpha0 = wu.deg2rad(90)
-    s = 1000000
-    P1_ana, alpha1_ana = gha1_ana(ell, P0, alpha0, s, maxM=60, maxPartCircum=32)
-    P1_app, alpha1_app, points = gha1_approx(ell, P0, alpha0, s, ds=5000, all_points=True)
-    show_points(points, P0, P1_ana)
-    print(np.linalg.norm(P1_app - P1_ana))
+    ell = EllipsoidTriaxial.init_name("KarneyTest2024")
+    P0 = ell.ell2cart(wu.deg2rad(15), wu.deg2rad(15))
+    alpha0 = wu.deg2rad(270)
+    s = 1
+    P1_app, alpha1_app, points, alphas = gha1_approx(ell, P0, alpha0, s, ds=0.1, all_points=True)
+    # P1_ana, alpha1_ana = gha1_ana(ell, P0, alpha0, s, maxM=40, maxPartCircum=32)
+    # print(np.linalg.norm(P1_app - P1_ana))
+    # show_points(points, P0, P0)

GHA_triaxial/gha1_num.py

@@ -0,0 +1,133 @@
+from typing import Callable, List, Tuple
+
+import numpy as np
+from numpy import arctan2, cos, sin
+from numpy.typing import NDArray
+
+import GHA_triaxial.numeric_examples_karney as ne_karney
+import runge_kutta as rk
+import winkelumrechnungen as wu
+from GHA_triaxial.gha1_ana import gha1_ana
+from GHA_triaxial.utils import alpha_ell2para, pq_ell
+from ellipsoid_triaxial import EllipsoidTriaxial
+from utils_angle import wrap_0_2pi
+
+
+def buildODE(ell: EllipsoidTriaxial) -> Callable:
+    """
+    Aufbau des DGL-Systems
+    :param ell: Ellipsoid
+    :return: DGL-System
+    """
+
+    def ODE(s: float, v: NDArray) -> NDArray:
+        """
+        DGL-System
+        :param s: unabhängige Variable
+        :param v: abhängige Variablen
+        :return: Ableitungen der abhängigen Variablen
+        """
+        x, dxds, y, dyds, z, dzds = v
+
+        H = ell.func_H(np.array([x, y, z]))
+        h = dxds ** 2 + 1 / (1 - ell.ee ** 2) * dyds ** 2 + 1 / (1 - ell.ex ** 2) * dzds ** 2
+
+        ddx = -(h / H) * x
+        ddy = -(h / H) * y / (1 - ell.ee ** 2)
+        ddz = -(h / H) * z / (1 - ell.ex ** 2)
+
+        return np.array([dxds, ddx, dyds, ddy, dzds, ddz])
+
+    return ODE
+
+def gha1_num(ell: EllipsoidTriaxial, point: NDArray, alpha0: float, s: float, num: int, all_points: bool = False) -> Tuple[NDArray, float] | Tuple[NDArray, float, List]:
+    """
+    Panou, Korakitits 2019
+    :param ell: Ellipsoid
+    :param point: Punkt in kartesischen Koordinaten
+    :param alpha0: Azimut im Startpunkt
+    :param s: Strecke
+    :param num: Anzahl Zwischenpunkte
+    :param all_points: Ausgabe aller Punkte?
+    :return: Zielpunkt, Azimut im Zielpunkt (, alle Punkte)
+    """
+    phi, lam, _ = ell.cart2geod(point, "ligas3")
+    p0 = ell.geod2cart(phi, lam, 0)
+    x0, y0, z0 = p0
+
+    p, q = pq_ell(ell, p0)
+    dxds0 = p[0] * sin(alpha0) + q[0] * cos(alpha0)
+    dyds0 = p[1] * sin(alpha0) + q[1] * cos(alpha0)
+    dzds0 = p[2] * sin(alpha0) + q[2] * cos(alpha0)
+
+    v_init = np.array([x0, dxds0, y0, dyds0, z0, dzds0])
+
+    ode = buildODE(ell)
+
+    _, werte = rk.rk4(ode, 0, v_init, s, num)
+    x1, dx1ds, y1, dy1ds, z1, dz1ds = werte[-1]
+
+    point1 = np.array([x1, y1, z1])
+
+    p1, q1 = pq_ell(ell, point1)
+    sigma = np.array([dx1ds, dy1ds, dz1ds])
+    P = float(p1 @ sigma)
+    Q = float(q1 @ sigma)
+
+    alpha1 = arctan2(P, Q)
+
+    alpha1 = wrap_0_2pi(alpha1)
+
+    _, _, h = ell.cart2geod(point1, "ligas3")
+    if h > 1e-5:
+        raise Exception("GHA1_num: explodiert, Punkt liegt nicht mehr auf dem Ellipsoid")
+
+    if all_points:
+        return point1, alpha1, werte
+    else:
+        return point1, alpha1
+
+if __name__ == "__main__":
+
+    # ell = ellipsoide.EllipsoidTriaxial.init_name("BursaSima1980round")
+    # diffs_panou = []
+    # examples_panou = ne_panou.get_random_examples(5)
+    # for example in examples_panou:
+    #     beta0, lamb0, beta1, lamb1, _, alpha0_ell, alpha1_ell, s = example
+    #     P0 = ell.ell2cart(beta0, lamb0)
+    #
+    #     P1_num, alpha1_num = gha1_num(ell, P0, alpha0_ell, s, 100)
+    #     beta1_num, lamb1_num = ell.cart2ell(P1_num)
+    #
+    #     _, _, alpha0_para = alpha_ell2para(ell, beta0, lamb0, alpha0_ell)
+    #     P1_ana, alpha1_ana = gha1_ana(ell, P0, alpha0_para, s, 60)
+    #     beta1_ana, lamb1_ana = ell.cart2ell(P1_ana)
+    #     diffs_panou.append((abs(beta1-beta1_num), abs(lamb1-lamb1_num), abs(beta1-beta1_ana), abs(lamb1-lamb1_ana)))
+    # diffs_panou = np.array(diffs_panou)
+    # mask_360 = (diffs_panou > 359) & (diffs_panou < 361)
+    # diffs_panou[mask_360] = np.abs(diffs_panou[mask_360] - 360)
+    # print(diffs_panou)
+
+    ell: EllipsoidTriaxial = EllipsoidTriaxial.init_name("KarneyTest2024")
+    diffs_karney = []
+    # examples_karney = ne_karney.get_examples((30499, 30500, 40500))
+    examples_karney = ne_karney.get_random_examples(20)
+    for example in examples_karney:
+        beta0, lamb0, alpha0_ell, beta1, lamb1, alpha1_ell, s = example
+        P0 = ell.ell2cart(beta0, lamb0)
+
+        P1_num, alpha1_num = gha1_num(ell, P0, alpha0_ell, s, 10000)
+        beta1_num, lamb1_num = ell.cart2ell(P1_num)
+
+        try:
+            _, _, alpha0_para = alpha_ell2para(ell, beta0, lamb0, alpha0_ell)
+            P1_ana, alpha1_ana = gha1_ana(ell, P0, alpha0_para, s, 45, maxPartCircum=32)
+            beta1_ana, lamb1_ana = ell.cart2ell(P1_ana)
+        except:
+            beta1_ana, lamb1_ana = np.inf, np.inf
+
+        diffs_karney.append((wu.rad2deg(abs(beta1-beta1_num)), wu.rad2deg(abs(lamb1-lamb1_num)), wu.rad2deg(abs(beta1-beta1_ana)), wu.rad2deg(abs(lamb1-lamb1_ana))))
+    diffs_karney = np.array(diffs_karney)
+    mask_360 = (diffs_karney > 359) & (diffs_karney < 361)
+    diffs_karney[mask_360] = np.abs(diffs_karney[mask_360] - 360)
+    print(diffs_karney)

GHA_triaxial/gha2_approx.py

@@ -1,15 +1,16 @@
-import numpy as np
-from ellipsoide import EllipsoidTriaxial
-from GHA_triaxial.panou_2013_2GHA_num import gha2_num
-import plotly.graph_objects as go
-import winkelumrechnungen as wu
-from numpy.typing import NDArray
 from typing import Tuple
 
-from utils import sigma2alpha
+import numpy as np
+import plotly.graph_objects as go
+from numpy.typing import NDArray
+
+import winkelumrechnungen as wu
+from GHA_triaxial.gha2_num import gha2_num
+from GHA_triaxial.utils import sigma2alpha
+from ellipsoid_triaxial import EllipsoidTriaxial
 
 
-def gha2(ell: EllipsoidTriaxial, p0: NDArray, p1: NDArray, ds: float, all_points: bool = False) -> Tuple[float, float, float] | Tuple[float, float, float, NDArray]:
+def gha2_approx(ell: EllipsoidTriaxial, p0: NDArray, p1: NDArray, ds: float, all_points: bool = False) -> Tuple[float, float, float] | Tuple[float, float, float, NDArray]:
     """
     Numerische Approximation für die zweite Hauptaufgabe
     :param ell: Ellipsoid
@@ -83,16 +84,16 @@ def show_points(points: NDArray, points_app: NDArray, p0: NDArray, p1: NDArray):
 
 
 if __name__ == '__main__':
-    ell = EllipsoidTriaxial.init_name("KarneyTest2024")
+    ell = EllipsoidTriaxial.init_name("Bursa1970")
 
-    beta0, lamb0 = (0.2, 0.1)
+    beta0, lamb0 = (0.1, 0.1)
     P0 = ell.ell2cart(beta0, lamb0)
-    beta1, lamb1 = (0.7, 0.3)
+    beta1, lamb1 = (0.3, np.pi)
     P1 = ell.ell2cart(beta1, lamb1)
 
-    alpha0_app, alpha1_app, s_app, points = gha2(ell, P0, P1, ds=1e-4, all_points=True)
-
-    alpha0, alpha1, s, betas, lambs = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=5000, all_points=True)
+    alpha0_app, alpha1_app, s_app, points = gha2_approx(ell, P0, P1, ds=100, all_points=True)
+    print("done")
+    alpha0, alpha1, s, betas, lambs = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=10000, all_points=True)
     points_ana = []
     for beta, lamb in zip(betas, lambs):
         points_ana.append(ell.ell2cart(beta, lamb))

GHA_triaxial/gha2_num.py

@@ -0,0 +1,643 @@
+from typing import Tuple
+
+import numpy as np
+from numpy.typing import NDArray
+
+import GHA_triaxial.numeric_examples_karney as ne_karney
+import GHA_triaxial.numeric_examples_panou as ne_panou
+import ausgaben as aus
+import winkelumrechnungen as wu
+from ellipsoid_triaxial import EllipsoidTriaxial
+from runge_kutta import rk4, rk4_end, rk4_integral
+from utils_angle import cot, wrap_0_2pi, wrap_mpi_pi
+
+
+def norm_a(a: float) -> float:
+    a = float(a) % (2 * np.pi)
+    return a
+
+
+def azimut(E: float, G: float, dbeta_du: float, dlamb_du: float) -> float:
+    north = np.sqrt(E) * dbeta_du
+    east = np.sqrt(G) * dlamb_du
+    return norm_a(np.arctan2(east, north))
+
+
+def sph_azimuth(beta1, lam1, beta2, lam2):
+    dlam = wrap_mpi_pi(lam2 - lam1)
+    y = np.sin(dlam) * np.cos(beta2)
+    x = np.cos(beta1) * np.sin(beta2) - np.sin(beta1) * np.cos(beta2) * np.cos(dlam)
+    a = np.arctan2(y, x)
+    if a < 0:
+        a += 2 * np.pi
+    return a
+
+
+# Panou 2013
+def gha2_num(
+    ell: EllipsoidTriaxial,
+    beta_0: float,
+    lamb_0: float,
+    beta_1: float,
+    lamb_1: float,
+    n: int = 16000,
+    epsilon: float = 10**-12,
+    iter_max: int = 30,
+    all_points: bool = False,
+) -> Tuple[float, float, float] | Tuple[float, float, float, NDArray, NDArray]:
+    """
+    :param ell: Ellipsoid
+    :param beta_0: Beta Punkt 0
+    :param lamb_0: Lambda Punkt 0
+    :param beta_1: Beta Punkt 1
+    :param lamb_1: Lambda Punkt 1
+    :param n: Anzahl Schritte
+    :param epsilon: Genauigkeit
+    :param iter_max: Maximale Iterationen
+    :param all_points: Ausgabe aller Punkte
+    :return: Azimut Startpunkt, Azumit Zielpunkt, Strecke
+    """
+
+    ax2 = float(ell.ax) * float(ell.ax)
+    ay2 = float(ell.ay) * float(ell.ay)
+    b2 = float(ell.b) * float(ell.b)
+    Ex2 = float(ell.Ex) * float(ell.Ex)
+    Ey2 = float(ell.Ey) * float(ell.Ey)
+    Ee2 = float(ell.Ee) * float(ell.Ee)
+    Ey4 = Ey2 * Ey2
+    Ee4 = Ee2 * Ee2
+    two_pi = 2.0 * np.pi
+
+    # Berechnung Koeffizienten, Gaußschen Fundamentalgrößen 1. Ordnung sowie deren Ableitungen
+    def BETA_LAMBDA(beta, lamb):
+        sb = np.sin(beta)
+        cb = np.cos(beta)
+        sl = np.sin(lamb)
+        cl = np.cos(lamb)
+
+        sb2 = sb * sb
+        cb2 = cb * cb
+        sl2 = sl * sl
+        cl2 = cl * cl
+
+        s2b = 2.0 * sb * cb
+        c2b = cb2 - sb2
+        s2l = 2.0 * sl * cl
+        c2l = cl2 - sl2
+
+        denB = Ex2 - Ey2 * sb2
+        denL = Ex2 - Ee2 * cl2
+
+        BETA = (ay2 * sb2 + b2 * cb2) / denB
+        LAMBDA = (ax2 * sl2 + ay2 * cl2) / denL
+
+        BETA_ = (ax2 * Ey2 * s2b) / (denB * denB)
+        LAMBDA_ = -(b2 * Ee2 * s2l) / (denL * denL)
+
+        BETA__ = (2.0 * ax2 * Ey4 * (s2b * s2b)) / (denB * denB * denB) + (2.0 * ax2 * Ey2 * c2b) / (
+            denB * denB
+        )
+        LAMBDA__ = (2.0 * b2 * Ee4 * (s2l * s2l)) / (denL * denL * denL) - (2.0 * b2 * Ee2 * s2l) / (
+            denL * denL
+        )
+
+        Q = Ey2 * cb2 + Ee2 * sl2
+
+        E = BETA * Q
+        G = LAMBDA * Q
+
+        E_beta = BETA_ * Q - BETA * Ey2 * s2b
+        E_lamb = BETA * Ee2 * s2l
+
+        G_beta = -LAMBDA * Ey2 * s2b
+        G_lamb = LAMBDA_ * Q + LAMBDA * Ee2 * s2l
+
+        E_beta_beta = BETA__ * Q - 2.0 * BETA_ * Ey2 * s2b - 2.0 * BETA * Ey2 * c2b
+        E_beta_lamb = BETA_ * Ee2 * s2l
+        E_lamb_lamb = 2.0 * BETA * Ee2 * c2l
+
+        G_beta_beta = -2.0 * LAMBDA * Ey2 * c2b
+        G_beta_lamb = -LAMBDA_ * Ey2 * s2b
+        G_lamb_lamb = LAMBDA__ * Q + 2.0 * LAMBDA_ * Ee2 * s2l + 2.0 * LAMBDA * Ee2 * c2l
+
+        return (
+            BETA,
+            LAMBDA,
+            E,
+            G,
+            BETA_,
+            LAMBDA_,
+            BETA__,
+            LAMBDA__,
+            E_beta,
+            E_lamb,
+            G_beta,
+            G_lamb,
+            E_beta_beta,
+            E_beta_lamb,
+            E_lamb_lamb,
+            G_beta_beta,
+            G_beta_lamb,
+            G_lamb_lamb,
+        )
+
+    # Berechnung der ODE Koeffizienten für Fall 1 (lambda_0 != lambda_1)
+    def p_coef(beta, lamb):
+        (
+            BETA,
+            LAMBDA,
+            E,
+            G,
+            BETA_,
+            LAMBDA_,
+            BETA__,
+            LAMBDA__,
+            E_beta,
+            E_lamb,
+            G_beta,
+            G_lamb,
+            E_beta_beta,
+            E_beta_lamb,
+            E_lamb_lamb,
+            G_beta_beta,
+            G_beta_lamb,
+            G_lamb_lamb,
+        ) = BETA_LAMBDA(beta, lamb)
+
+        p_3 = -0.5 * (E_lamb / G)
+        p_2 = (G_beta / G) - 0.5 * (E_beta / E)
+        p_1 = 0.5 * (G_lamb / G) - (E_lamb / E)
+        p_0 = 0.5 * (G_beta / E)
+
+        p_33 = -0.5 * ((E_beta_lamb * G - E_lamb * G_beta) / (G**2))
+        p_22 = ((G * G_beta_beta - G_beta * G_beta) / (G**2)) - 0.5 * (
+            (E * E_beta_beta - E_beta * E_beta) / (E**2)
+        )
+        p_11 = 0.5 * ((G * G_beta_lamb - G_beta * G_lamb) / (G**2)) - (
+            (E * E_beta_lamb - E_beta * E_lamb) / (E**2)
+        )
+        p_00 = 0.5 * ((E * G_beta_beta - E_beta * G_beta) / (E**2))
+
+        return BETA, LAMBDA, E, G, p_3, p_2, p_1, p_0, p_33, p_22, p_11, p_00
+
+    # Berechnung der ODE Koeffizienten für Fall 2 (lambda_0 == lambda_1)
+    def q_coef(beta, lamb):
+        (
+            BETA,
+            LAMBDA,
+            E,
+            G,
+            BETA_,
+            LAMBDA_,
+            BETA__,
+            LAMBDA__,
+            E_beta,
+            E_lamb,
+            G_beta,
+            G_lamb,
+            E_beta_beta,
+            E_beta_lamb,
+            E_lamb_lamb,
+            G_beta_beta,
+            G_beta_lamb,
+            G_lamb_lamb,
+        ) = BETA_LAMBDA(beta, lamb)
+
+        q_3 = -0.5 * (G_beta / E)
+        q_2 = (E_lamb / E) - 0.5 * (G_lamb / G)
+        q_1 = 0.5 * (E_beta / E) - (G_beta / G)
+        q_0 = 0.5 * (E_lamb / G)
+
+        q_33 = -0.5 * ((E * G_beta_lamb - E_lamb * G_lamb) / (E**2))
+        q_22 = ((E * E_lamb_lamb - E_lamb * E_lamb) / (E**2)) - 0.5 * (
+            (G * G_lamb_lamb - G_lamb * G_lamb) / (G**2)
+        )
+        q_11 = 0.5 * ((E * E_beta_lamb - E_beta * E_lamb) / (E**2)) - (
+            (G * G_beta_lamb - G_beta * G_lamb) / (G**2)
+        )
+        q_00 = 0.5 * ((E_lamb_lamb * G - E_lamb * G_lamb) / (G**2))
+
+        return BETA, LAMBDA, E, G, q_3, q_2, q_1, q_0, q_33, q_22, q_11, q_00
+
+    def integrand_lambda(lamb, y):
+        beta = y[0]
+        beta_p = y[1]
+        (_, _, E, G, *_) = BETA_LAMBDA(beta, lamb)
+        return np.sqrt(E * beta_p**2 + G)
+
+    def integrand_beta(beta, y):
+        lamb = y[0]
+        lamb_p = y[1]
+        (_, _, E, G, *_) = BETA_LAMBDA(beta, lamb)
+        return np.sqrt(E + G * lamb_p**2)
+
+    def solve_lambda_branch(beta0, lamb0, beta1, lamb1_target, N_run, N_newt, it_max):
+        dlamb = float(lamb1_target - lamb0)
+        if abs(dlamb) < 1e-15:
+            return None
+
+        sgn = 1.0 if dlamb >= 0.0 else -1.0
+
+        def ode_lamb(lamb, v):
+            beta, beta_p, X3, X4 = v
+            (_, _, _, _, p_3, p_2, p_1, p_0, p_33, p_22, p_11, p_00) = p_coef(beta, lamb)
+
+            dbeta = beta_p
+            dbeta_p = p_3 * beta_p**3 + p_2 * beta_p**2 + p_1 * beta_p + p_0
+            dX3 = X4
+            dX4 = (p_33 * beta_p**3 + p_22 * beta_p**2 + p_11 * beta_p + p_00) * X3 + (
+                3 * p_3 * beta_p**2 + 2 * p_2 * beta_p + p_1
+            ) * X4
+            return np.array([dbeta, dbeta_p, dX3, dX4], dtype=float)
+
+        alpha0_sph = sph_azimuth(beta0, lamb0, beta1, lamb1_target)
+        (_, _, E0, G0, *_) = BETA_LAMBDA(beta0, lamb0)
+        beta_p0_sph = np.sqrt(G0 / E0) * cot(alpha0_sph)
+
+        def solve_newton(beta_p0_init: float):
+            beta_p0 = float(beta_p0_init)
+            for _ in range(it_max):
+                v0 = np.array([beta0, beta_p0, 0.0, 1.0], dtype=float)
+                _, y_end = rk4_end(ode_lamb, lamb0, v0, dlamb, N_newt)
+
+                beta_end, _, X3_end, _ = y_end
+                delta = beta_end - beta1
+
+                if abs(delta) < epsilon:
+                    return True, beta_p0
+
+                if abs(X3_end) < 1e-20:
+                    return False, None
+
+                step = delta / X3_end
+                step = float(np.clip(step, -0.5, 0.5))
+                beta_p0 -= step
+
+            return False, None
+
+        seeds = [beta_p0_sph, -beta_p0_sph, 0.5 * beta_p0_sph, 2.0 * beta_p0_sph]
+        best = None
+
+        for seed in seeds:
+            ok, sol = solve_newton(seed)
+            if not ok:
+                continue
+            v0_sol = np.array([beta0, sol, 0.0, 1.0], dtype=float)
+            _, _, s_val = rk4_integral(ode_lamb, lamb0, v0_sol, dlamb, N_run, integrand_lambda)
+            if (best is None) or (s_val < best[0]):
+                best = (float(s_val), float(sol))
+
+        if best is None:
+            return None
+
+        return best[0], best[1], sgn, dlamb, ode_lamb
+
+    def solve_beta_branch(beta0, lamb0, beta1, lamb1, N_run, N_newt, it_max):
+        dbeta = float(beta1 - beta0)
+        if abs(dbeta) < 1e-15:
+            return None
+
+        sgn = 1.0 if dbeta >= 0.0 else -1.0
+
+        def ode_beta(beta, v):
+            lamb, lamb_p, Y3, Y4 = v
+            (_, _, _, _, q_3, q_2, q_1, q_0, q_33, q_22, q_11, q_00) = q_coef(beta, lamb)
+
+            dlamb = lamb_p
+            dlamb_p = q_3 * lamb_p**3 + q_2 * lamb_p**2 + q_1 * lamb_p + q_0
+            dY3 = Y4
+            dY4 = (q_33 * lamb_p**3 + q_22 * lamb_p**2 + q_11 * lamb_p + q_00) * Y3 + (
+                3 * q_3 * lamb_p**2 + 2 * q_2 * lamb_p + q_1
+            ) * Y4
+            return np.array([dlamb, dlamb_p, dY3, dY4], dtype=float)
+
+        def solve_newton(lamb_p0_init: float):
+            lamb_p0 = float(lamb_p0_init)
+            for _ in range(it_max):
+                v0 = np.array([lamb0, lamb_p0, 0.0, 1.0], dtype=float)
+                _, y_end = rk4_end(ode_beta, beta0, v0, dbeta, N_newt)
+
+                lamb_end, _, Y3_end, _ = y_end
+                delta = lamb_end - lamb1
+
+                if abs(delta) < epsilon:
+                    return True, lamb_p0
+
+                if abs(Y3_end) < 1e-20:
+                    return False, None
+
+                step = delta / Y3_end
+                step = float(np.clip(step, -1.0, 1.0))
+                lamb_p0 -= step
+
+            return False, None
+
+        seeds = [0.0, 0.25, -0.25, 1.0, -1.0]
+        best = None
+
+        for seed in seeds:
+            ok, sol = solve_newton(seed)
+            if not ok:
+                continue
+            v0_sol = np.array([lamb0, sol, 0.0, 1.0], dtype=float)
+            _, _, s_val = rk4_integral(ode_beta, beta0, v0_sol, dbeta, N_run, integrand_beta)
+            if (best is None) or (s_val < best[0]):
+                best = (float(s_val), float(sol))
+
+        if best is None:
+            return None
+
+        return best[0], best[1], sgn, dbeta, ode_beta
+
+    lamb0 = float(wrap_mpi_pi(lamb_0))
+    lamb1 = float(wrap_mpi_pi(lamb_1))
+    beta0 = float(beta_0)
+    beta1 = float(beta_1)
+
+    N_full = int(n)
+    if N_full < 2:
+        N_full = 2
+
+    if all_points:
+        N_fast = min(2000, max(400, N_full // 10))
+    else:
+        N_fast = min(1500, max(300, N_full // 12))
+
+    k0 = int(np.round((lamb0 - lamb1) / two_pi))
+    lamb_targets = []
+    for dk in (-1, 0, 1):
+        lt = lamb1 + two_pi * float(k0 + dk)
+        dl = lt - lamb0
+        if abs(dl) <= np.pi + 1e-12:
+            lamb_targets.append(float(lt))
+    if not lamb_targets:
+        lamb_targets = [float(lamb1 + two_pi * float(k0))]
+
+    best_fast = None
+
+    for lt in lamb_targets:
+        if abs(lt - lamb0) >= 1e-15:
+            res = solve_lambda_branch(beta0, lamb0, beta1, lt, N_fast, min(N_fast, 800), min(iter_max, 12))
+            if res is None:
+                continue
+            s_fast, beta_p0_fast, sgn_fast, dlamb_fast, _ = res
+            cand = ("lambda", s_fast, lt, beta_p0_fast, sgn_fast, dlamb_fast)
+        else:
+            res = solve_beta_branch(beta0, lamb0, beta1, lamb1, N_fast, min(N_fast, 800), min(iter_max, 12))
+            if res is None:
+                continue
+            s_fast, lamb_p0_fast, sgn_fast, dbeta_fast, _ = res
+            cand = ("beta", s_fast, lt, lamb_p0_fast, sgn_fast, dbeta_fast)
+
+        if (best_fast is None) or (cand[1] < best_fast[1]):
+            best_fast = cand
+
+    if best_fast is None:
+        if abs(lamb1 - lamb0) >= 1e-15:
+            best_fast = ("lambda", 0.0, lamb1, None, 1.0, float(lamb1 - lamb0))
+        else:
+            best_fast = ("beta", 0.0, lamb1, None, 1.0, float(beta1 - beta0))
+
+    if best_fast[0] == "lambda":
+        lt = float(best_fast[2])
+        dlamb = float(lt - lamb0)
+        sgn = 1.0 if dlamb >= 0.0 else -1.0
+
+        def ode_lamb(lamb, v):
+            beta, beta_p, X3, X4 = v
+            (_, _, _, _, p_3, p_2, p_1, p_0, p_33, p_22, p_11, p_00) = p_coef(beta, lamb)
+
+            dbeta = beta_p
+            dbeta_p = p_3 * beta_p**3 + p_2 * beta_p**2 + p_1 * beta_p + p_0
+            dX3 = X4
+            dX4 = (p_33 * beta_p**3 + p_22 * beta_p**2 + p_11 * beta_p + p_00) * X3 + (
+                3 * p_3 * beta_p**2 + 2 * p_2 * beta_p + p_1
+            ) * X4
+            return np.array([dbeta, dbeta_p, dX3, dX4], dtype=float)
+
+        alpha0_sph = sph_azimuth(beta0, lamb0, beta1, lt)
+        (_, _, E0, G0, *_) = BETA_LAMBDA(beta0, lamb0)
+        beta_p0_sph = np.sqrt(G0 / E0) * cot(alpha0_sph)
+
+        beta_p0_init = best_fast[3]
+        if beta_p0_init is None:
+            beta_p0_init = beta_p0_sph
+        beta_p0_init = float(beta_p0_init)
+
+        N_newton = min(N_full, 4000)
+
+        def solve_newton_refine(beta_p0_init: float):
+            beta_p0 = float(beta_p0_init)
+            for _ in range(iter_max):
+                v0 = np.array([beta0, beta_p0, 0.0, 1.0], dtype=float)
+                _, y_end = rk4_end(ode_lamb, lamb0, v0, dlamb, N_newton)
+
+                beta_end, _, X3_end, _ = y_end
+                delta = beta_end - beta1
+
+                if abs(delta) < epsilon:
+                    return True, beta_p0
+
+                if abs(X3_end) < 1e-20:
+                    return False, None
+
+                step = delta / X3_end
+                step = float(np.clip(step, -0.5, 0.5))
+                beta_p0 -= step
+
+            return False, None
+
+        ok, beta_p0_sol = solve_newton_refine(beta_p0_init)
+
+        if not ok:
+            seeds = [beta_p0_sph, -beta_p0_sph, 0.5 * beta_p0_sph, 2.0 * beta_p0_sph]
+            best = None
+            for seed in seeds:
+                ok_s, sol_s = solve_newton_refine(seed)
+                if not ok_s:
+                    continue
+                v0_s = np.array([beta0, sol_s, 0.0, 1.0], dtype=float)
+                _, _, s_s = rk4_integral(ode_lamb, lamb0, v0_s, dlamb, min(N_full, 2000), integrand_lambda)
+                if (best is None) or (s_s < best[0]):
+                    best = (float(s_s), float(sol_s))
+            if best is None:
+                raise RuntimeError("GHA2_num: Keine Startwert-Variante konvergiert (lambda-Fall)")
+            beta_p0_sol = best[1]
+
+        beta_p0 = float(beta_p0_sol)
+        v0_final = np.array([beta0, beta_p0, 0.0, 1.0], dtype=float)
+
+        if all_points:
+            lamb_list, states = rk4(ode_lamb, lamb0, v0_final, dlamb, N_full, False)
+            lamb_arr = np.array(lamb_list, dtype=float)
+            beta_arr = np.array([st[0] for st in states], dtype=float)
+            beta_p_arr = np.array([st[1] for st in states], dtype=float)
+
+            (_, _, E_start, G_start, *_) = BETA_LAMBDA(beta_arr[0], lamb_arr[0])
+            (_, _, E_end, G_end, *_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
+
+            alpha_0 = azimut(E_start, G_start, dbeta_du=beta_p_arr[0] * sgn, dlamb_du=1.0 * sgn)
+            alpha_1 = azimut(E_end, G_end, dbeta_du=beta_p_arr[-1] * sgn, dlamb_du=1.0 * sgn)
+
+            integrand = np.zeros(N_full + 1, dtype=float)
+            for i in range(N_full + 1):
+                (_, _, Ei, Gi, *_) = BETA_LAMBDA(beta_arr[i], lamb_arr[i])
+                integrand[i] = np.sqrt(Ei * beta_p_arr[i] ** 2 + Gi)
+
+            h = abs(dlamb) / N_full
+            if N_full % 2 == 0:
+                S = integrand[0] + integrand[-1] + 4.0 * np.sum(integrand[1:-1:2]) + 2.0 * np.sum(
+                    integrand[2:-1:2]
+                )
+                s = h / 3.0 * S
+            else:
+                s = np.trapz(integrand, dx=h)
+
+            return float(wrap_0_2pi(alpha_0)), float(wrap_0_2pi(alpha_1)), float(s), beta_arr, lamb_arr
+
+        _, y_end, s = rk4_integral(ode_lamb, lamb0, v0_final, dlamb, N_full, integrand_lambda)
+        beta_end, beta_p_end, _, _ = y_end
+
+        (_, _, E_start, G_start, *_) = BETA_LAMBDA(beta0, lamb0)
+        alpha_0 = azimut(E_start, G_start, dbeta_du=beta_p0 * sgn, dlamb_du=1.0 * sgn)
+
+        (_, _, E_end, G_end, *_) = BETA_LAMBDA(float(beta_end), float(lamb0 + dlamb))
+        alpha_1 = azimut(E_end, G_end, dbeta_du=float(beta_p_end) * sgn, dlamb_du=1.0 * sgn)
+
+        return float(wrap_0_2pi(alpha_0)), float(wrap_0_2pi(alpha_1)), float(s)
+
+    # Fall 2 (lambda_0 == lambda_1)
+    N = int(n)
+    dbeta = float(beta_1 - beta_0)
+
+    if abs(dbeta) < 1e-15:
+        if all_points:
+            return 0.0, 0.0, 0.0, np.array([]), np.array([])
+        return 0.0, 0.0, 0.0
+
+    sgn = 1.0 if dbeta >= 0.0 else -1.0
+
+    def ode_beta(beta, v):
+        lamb, lamb_p, Y3, Y4 = v
+        (_, _, _, _, q_3, q_2, q_1, q_0, q_33, q_22, q_11, q_00) = q_coef(beta, lamb)
+
+        dlamb = lamb_p
+        dlamb_p = q_3 * lamb_p**3 + q_2 * lamb_p**2 + q_1 * lamb_p + q_0
+        dY3 = Y4
+        dY4 = (q_33 * lamb_p**3 + q_22 * lamb_p**2 + q_11 * lamb_p + q_00) * Y3 + (
+            3 * q_3 * lamb_p**2 + 2 * q_2 * lamb_p + q_1
+        ) * Y4
+        return np.array([dlamb, dlamb_p, dY3, dY4], dtype=float)
+
+    lamb_p0 = float(best_fast[3]) if (best_fast[0] == "beta" and best_fast[3] is not None) else 0.0
+
+    for _ in range(iter_max):
+        v0 = np.array([lamb0, lamb_p0, 0.0, 1.0], dtype=float)
+        _, y_end = rk4_end(ode_beta, beta0, v0, dbeta, N)
+
+        lamb_end, _, Y3_end, _ = y_end
+        delta = lamb_end - lamb1
+
+        if abs(delta) < epsilon:
+            break
+
+        if abs(Y3_end) < 1e-20:
+            raise RuntimeError("GHA2_num: Ableitung ~ 0 im beta-Fall")
+
+        step = delta / Y3_end
+        step = float(np.clip(step, -1.0, 1.0))
+        lamb_p0 -= step
+
+    v0_final = np.array([lamb0, lamb_p0, 0.0, 1.0], dtype=float)
+
+    if all_points:
+        beta_list, states = rk4(ode_beta, beta0, v0_final, dbeta, N, False)
+        beta_arr = np.array(beta_list, dtype=float)
+        lamb_arr = np.array([st[0] for st in states], dtype=float)
+        lamb_p_arr = np.array([st[1] for st in states], dtype=float)
+
+        (_, _, E_start, G_start, *_) = BETA_LAMBDA(beta_arr[0], lamb_arr[0])
+        (_, _, E_end, G_end, *_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
+
+        alpha_0 = azimut(E_start, G_start, dbeta_du=1.0 * sgn, dlamb_du=lamb_p_arr[0] * sgn)
+        alpha_1 = azimut(E_end, G_end, dbeta_du=1.0 * sgn, dlamb_du=lamb_p_arr[-1] * sgn)
+
+        integrand = np.zeros(N + 1, dtype=float)
+        for i in range(N + 1):
+            (_, _, Ei, Gi, *_) = BETA_LAMBDA(beta_arr[i], lamb_arr[i])
+            integrand[i] = np.sqrt(Ei + Gi * lamb_p_arr[i] ** 2)
+
+        h = abs(dbeta) / N
+        if N % 2 == 0:
+            S = integrand[0] + integrand[-1] + 4.0 * np.sum(integrand[1:-1:2]) + 2.0 * np.sum(
+                integrand[2:-1:2]
+            )
+            s = h / 3.0 * S
+        else:
+            s = np.trapz(integrand, dx=h)
+
+        return float(wrap_0_2pi(alpha_0)), float(wrap_0_2pi(alpha_1)), float(s), beta_arr, lamb_arr
+
+    _, y_end, s = rk4_integral(ode_beta, beta0, v0_final, dbeta, N, integrand_beta)
+    lamb_end, lamb_p_end, _, _ = y_end
+
+    (_, _, E_start, G_start, *_) = BETA_LAMBDA(beta0, lamb0)
+    alpha_0 = azimut(E_start, G_start, dbeta_du=1.0 * sgn, dlamb_du=lamb_p0 * sgn)
+
+    (_, _, E_end, G_end, *_) = BETA_LAMBDA(beta1, float(lamb_end))
+    alpha_1 = azimut(E_end, G_end, dbeta_du=1.0 * sgn, dlamb_du=float(lamb_p_end) * sgn)
+
+    return float(wrap_0_2pi(alpha_0)), float(wrap_0_2pi(alpha_1)), float(s)
+
+
+if __name__ == "__main__":
+    ell = EllipsoidTriaxial.init_name("BursaSima1980round")
+    beta1 = np.deg2rad(75)
+    lamb1 = np.deg2rad(-90)
+    beta2 = np.deg2rad(75)
+    lamb2 = np.deg2rad(66)
+    a0, a1, s = gha2_num(ell, beta1, lamb1, beta2, lamb2, n=100)
+    print(aus.gms("a0", a0, 4))
+    print(aus.gms("a1", a1, 4))
+    print("s: ", s)
+    # print(aus.gms("a2", a2, 4))
+    # print(s)
+    cart1 = ell.para2cart(0, 0)
+    cart2 = ell.para2cart(0.4, 1.4)
+    beta1, lamb1 = ell.cart2ell(cart1)
+    beta2, lamb2 = ell.cart2ell(cart2)
+
+    a1, a2, s = gha2_num(ell, beta1, lamb1, beta2, lamb2, n=5000)
+    print(s)
+
+    ell = EllipsoidTriaxial.init_name("BursaSima1980round")
+    diffs_panou = []
+    examples_panou = ne_panou.get_random_examples(4)
+    for example in examples_panou:
+        beta0, lamb0, beta1, lamb1, _, alpha0, alpha1, s = example
+        P0 = ell.ell2cart(beta0, lamb0)
+        try:
+            alpha0_num, alpha1_num, s_num = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=4000, iter_max=10)
+            diffs_panou.append(
+                (wu.rad2deg(abs(alpha0 - alpha0_num)), wu.rad2deg(abs(alpha1 - alpha1_num)), abs(s - s_num)))
+        except:
+            print(f"Fehler für {beta0}, {lamb0}, {beta1}, {lamb1}")
+    diffs_panou = np.array(diffs_panou)
+    print(diffs_panou)
+
+    ell = EllipsoidTriaxial.init_name("KarneyTest2024")
+    diffs_karney = []
+    # examples_karney = ne_karney.get_examples((30500, 40500))
+    examples_karney = ne_karney.get_random_examples(2)
+    for example in examples_karney:
+        beta0, lamb0, alpha0, beta1, lamb1, alpha1, s = example
+
+        try:
+            alpha0_num, alpha1_num, s_num = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=4000, iter_max=10)
+            diffs_karney.append((wu.rad2deg(abs(alpha0-alpha0_num)), wu.rad2deg(abs(alpha1-alpha1_num)), abs(s-s_num)))
+        except:
+            print(f"Fehler für {beta0}, {lamb0}, {beta1}, {lamb1}")
+    diffs_karney = np.array(diffs_karney)
+    print(diffs_karney)
+
+    pass

GHA_triaxial/numeric_examples_karney.py

@@ -1,6 +1,12 @@
 import random
+from typing import List
+
 import winkelumrechnungen as wu
-from typing import List, Tuple
+from GHA_triaxial.utils import jacobi_konstante
+from ellipsoid_triaxial import EllipsoidTriaxial
+
+ell = EllipsoidTriaxial.init_name("KarneyTest2024")
+file_path = r"Karney_2024_Testset.txt"
 
 def line2example(line: str) -> List:
     """
@@ -26,7 +32,7 @@ def get_random_examples(num: int, seed: int = None) -> List:
     """
     if seed is not None:
         random.seed(seed)
-    with open("Karney_2024_Testset.txt") as datei:
+    with open(file_path) as datei:
         lines = datei.readlines()
     examples = []
     for i in range(num):
@@ -41,7 +47,7 @@ def get_examples(l_i: List) -> List:
     :param l_i: Liste von Indizes
     :return: Liste mit Beispielen
     """
-    with open("Karney_2024_Testset.txt") as datei:
+    with open(file_path) as datei:
         lines = datei.readlines()
     examples = []
     for i in l_i:
@@ -50,5 +56,63 @@ def get_examples(l_i: List) -> List:
     return examples
 
 
+def get_random_examples_gamma(group: str, num: int, seed: int = None, length: str = None) -> List:
+    """
+    Zufällige Beispiele aus Karney in Gruppen nach Einteilung anhand der Jacobi-Konstanten
+    :param group: Gruppe
+    :param num: Anzahl
+    :param seed: Random-Seed
+    :param length: long oder short, sond egal
+    :return: Liste mit Beispielen
+    """
+    eps = 1e-20
+    long_short = 2
+    if seed is not None:
+        random.seed(seed)
+    with open(file_path) as datei:
+        lines = datei.readlines()
+    examples = []
+    i = 0
+    while len(examples) < num and i < len(lines):
+        example = line2example(lines[random.randint(0, len(lines) - 1)])
+        if example in examples:
+            continue
+        i += 1
+
+        beta0, lamb0, alpha0_ell, beta1, lamb1, alpha1_ell, s = example
+        gamma = jacobi_konstante(beta0, lamb0, alpha0_ell, ell)
+
+        if group not in ["a", "b", "c", "d", "e", "de"]:
+            break
+        elif group == "a" and not 1 >= gamma >= 0.01:
+            continue
+        elif group == "b" and not 0.01 > gamma > eps:
+            continue
+        elif group == "c" and not abs(gamma) <= eps:
+            continue
+        elif group == "d" and not -eps > gamma > -1e-17:
+            continue
+        elif group == "e" and not -1e-17 >= gamma >= -1:
+            continue
+        elif group == "de" and not -eps > gamma > -1:
+            continue
+
+        if length == "short":
+            if example[6] < long_short:
+                examples.append(example)
+        elif length == "long":
+            if example[6] >= long_short:
+                examples.append(example)
+        else:
+            examples.append(example)
+
+    return examples
+
+
 if __name__ == "__main__":
-    get_random_examples(10)
+    examples_a = get_random_examples_gamma("a", 10, 42)
+    examples_b = get_random_examples_gamma("b", 10, 42)
+    examples_c = get_random_examples_gamma("c", 10, 42)
+    examples_d = get_random_examples_gamma("d", 10, 42)
+    examples_e = get_random_examples_gamma("e", 10, 42)
+    pass

GHA_triaxial/numeric_examples_panou.py

@@ -72,33 +72,33 @@ table3 = [
      wu.gms2rad([0, -1, 27.9705]),     wu.gms2rad([0, 0, 16.0490]),       8888783.7815)
 ]
 
-table4 = [
-    (wu.deg2rad(0), wu.deg2rad(0),     wu.deg2rad(0), wu.deg2rad(90),     1.00000000000,
-     wu.gms2rad([90, 0, 0.0000]),      wu.gms2rad([90, 0, 0.0000]),       10018754.1714),
+# table4 = [
+#     (wu.deg2rad(0), wu.deg2rad(0),     wu.deg2rad(0), wu.deg2rad(90),     1.00000000000,
+#      wu.gms2rad([90, 0, 0.0000]),      wu.gms2rad([90, 0, 0.0000]),       10018754.1714),
+#
+#     (wu.deg2rad(1), wu.deg2rad(0),     wu.deg2rad(0), wu.deg2rad(179.5),  0.30320665822,
+#      wu.gms2rad([17, 39, 11.0942]),    wu.gms2rad([162, 20, 58.9032]),    19884417.8083),
+#
+#     (wu.deg2rad(5), wu.deg2rad(0),     wu.deg2rad(-80), wu.deg2rad(170),  0.03104258442,
+#      wu.gms2rad([178, 12, 51.5083]),   wu.gms2rad([10, 17, 52.6423]),     11652530.7514),
+#
+#     (wu.deg2rad(30), wu.deg2rad(0),    wu.deg2rad(-75), wu.deg2rad(120),  0.24135347134,
+#      wu.gms2rad([163, 49, 4.4615]),    wu.gms2rad([68, 49, 50.9617]),     14057886.8752),
+#
+#     (wu.deg2rad(60), wu.deg2rad(0),    wu.deg2rad(-60), wu.deg2rad(40),   0.19408499032,
+#      wu.gms2rad([157, 9, 33.5589]),    wu.gms2rad([157, 9, 33.5589]),     13767414.8267),
+#
+#     (wu.deg2rad(75), wu.deg2rad(0),    wu.deg2rad(-30), wu.deg2rad(0.5),  0.00202789418,
+#      wu.gms2rad([179, 33, 3.8613]),    wu.gms2rad([179, 51, 57.0077]),    11661713.4496),
+#
+#     (wu.deg2rad(80), wu.deg2rad(0),    wu.deg2rad(-5), wu.deg2rad(120),   0.15201222384,
+#      wu.gms2rad([61, 5, 33.9600]),     wu.gms2rad([171, 13, 22.0148]),    11105138.2902),
+#
+#     (wu.deg2rad(0), wu.deg2rad(0),     wu.deg2rad(60), wu.deg2rad(0),     0.00000000000,
+#      wu.gms2rad([0, 0, 0.0000]),       wu.gms2rad([0, 0, 0.0000]),     6663348.2060)
+# ]
 
-    (wu.deg2rad(1), wu.deg2rad(0),     wu.deg2rad(0), wu.deg2rad(179.5),  0.30320665822,
-     wu.gms2rad([17, 39, 11.0942]),    wu.gms2rad([162, 20, 58.9032]),    19884417.8083),
-
-    (wu.deg2rad(5), wu.deg2rad(0),     wu.deg2rad(-80), wu.deg2rad(170),  0.03104258442,
-     wu.gms2rad([178, 12, 51.5083]),   wu.gms2rad([10, 17, 52.6423]),     11652530.7514),
-
-    (wu.deg2rad(30), wu.deg2rad(0),    wu.deg2rad(-75), wu.deg2rad(120),  0.24135347134,
-     wu.gms2rad([163, 49, 4.4615]),    wu.gms2rad([68, 49, 50.9617]),     14057886.8752),
-
-    (wu.deg2rad(60), wu.deg2rad(0),    wu.deg2rad(-60), wu.deg2rad(40),   0.19408499032,
-     wu.gms2rad([157, 9, 33.5589]),    wu.gms2rad([157, 9, 33.5589]),     13767414.8267),
-
-    (wu.deg2rad(75), wu.deg2rad(0),    wu.deg2rad(-30), wu.deg2rad(0.5),  0.00202789418,
-     wu.gms2rad([179, 33, 3.8613]),    wu.gms2rad([179, 51, 57.0077]),    11661713.4496),
-
-    (wu.deg2rad(80), wu.deg2rad(0),    wu.deg2rad(-5), wu.deg2rad(120),   0.15201222384,
-     wu.gms2rad([61, 5, 33.9600]),     wu.gms2rad([171, 13, 22.0148]),    11105138.2902),
-
-    (wu.deg2rad(0), wu.deg2rad(0),     wu.deg2rad(60), wu.deg2rad(0),     0.00000000000,
-     wu.gms2rad([0, 0, 0.0000]),       wu.gms2rad([0, 0, 0.0000]),     6663348.2060)
-]
-
-tables = [table1, table2, table3, table4]
+tables = [table1, table2, table3]
 
 def get_example(table: int, example: int) -> Tuple:
     """
@@ -110,18 +110,25 @@ def get_example(table: int, example: int) -> Tuple:
     table -= 1
     example -= 1
     tables = get_tables()
-    return tables[table][example]
+    beta0, lamb0, beta1, lamb1, _, alpha0_ell, alpha1_ell, s = tables[table][example]
+    return beta0, lamb0, alpha0_ell, beta1, lamb1, alpha1_ell, s
 
 def get_tables() -> List:
     """
     Rückgabe aller Tabellen
     :return: Alle Tabellen
     """
-    return tables
+    sorted_tables = []
+    for table in tables:
+        sorted_tables.append([])
+        for example in table:
+            beta0, lamb0, beta1, lamb1, _, alpha0_ell, alpha1_ell, s = example
+            sorted_tables[-1].append((beta0, lamb0, alpha0_ell, beta1, lamb1, alpha1_ell, s))
+    return sorted_tables
 
 def get_random_examples(num: int, seed: int = None) -> List:
     """
-    Rückgabe zufäliger Beispiele
+    Rückgabe zufälliger Beispiele
     :param num: Anzahl Beispiele
     :param seed: Random-Seed
     :return:
@@ -131,7 +138,7 @@ def get_random_examples(num: int, seed: int = None) -> List:
 
     examples = []
     for i in range(num):
-        table = random.randint(1, 4)
+        table = random.randint(1, 3)
         if table == 4:
             example = random.randint(1, 8)
         else:

GHA_triaxial/panou.py

@@ -1,360 +0,0 @@
-import numpy as np
-from numpy import sin, cos, sqrt, arctan2
-import ellipsoide
-import runge_kutta as rk
-import winkelumrechnungen as wu
-from scipy.special import factorial as fact
-from math import comb
-import GHA_triaxial.numeric_examples_panou as ne_panou
-import GHA_triaxial.numeric_examples_karney as ne_karney
-from ellipsoide import EllipsoidTriaxial
-from typing import Callable, Tuple, List
-from numpy.typing import NDArray
-
-
-def pq_ell(ell: EllipsoidTriaxial, point: NDArray) -> Tuple[NDArray, NDArray]:
-    """
-    Berechnung von p und q in elliptischen Koordinaten
-    Panou, Korakitits 2019
-    :param ell: Ellipsoid
-    :param point: Punkt
-    :return: p und q
-    """
-    x, y, z = point
-    n = ell.func_n(point)
-
-    beta, lamb = ell.cart2ell(point)
-    B = ell.Ex ** 2 * cos(beta) ** 2 + ell.Ee ** 2 * sin(beta) ** 2
-    L = ell.Ex ** 2 - ell.Ee ** 2 * cos(lamb) ** 2
-
-    c1 = x ** 2 + y ** 2 + z ** 2 - (ell.ax ** 2 + ell.ay ** 2 + ell.b ** 2)
-    c0 = (ell.ax ** 2 * ell.ay ** 2 + ell.ax ** 2 * ell.b ** 2 + ell.ay ** 2 * ell.b ** 2 -
-          (ell.ay ** 2 + ell.b ** 2) * x ** 2 - (ell.ax ** 2 + ell.b ** 2) * y ** 2 - (
-                  ell.ax ** 2 + ell.ay ** 2) * z ** 2)
-    t2 = (-c1 + sqrt(c1 ** 2 - 4 * c0)) / 2
-
-    F = ell.Ey ** 2 * cos(beta) ** 2 + ell.Ee ** 2 * sin(lamb) ** 2
-    p1 = -sqrt(L / (F * t2)) * ell.ax / ell.Ex * sqrt(B) * sin(lamb)
-    p2 = sqrt(L / (F * t2)) * ell.ay * cos(beta) * cos(lamb)
-    p3 = 1 / sqrt(F * t2) * (ell.b * ell.Ee ** 2) / (2 * ell.Ex) * sin(beta) * sin(2 * lamb)
-    p = np.array([p1, p2, p3])
-    q = np.array([n[1] * p[2] - n[2] * p[1],
-                  n[2] * p[0] - n[0] * p[2],
-                  n[0] * p[1] - n[1] * p[0]])
-
-    return p, q
-
-def buildODE(ell: EllipsoidTriaxial) -> Callable:
-    """
-    Aufbau des DGL-Systems
-    :param ell: Ellipsoid
-    :return: DGL-System
-    """
-    def ODE(s: float, v: NDArray) -> NDArray:
-        """
-        DGL-System
-        :param s: unabhängige Variable
-        :param v: abhängige Variablen
-        :return: Ableitungen der abhängigen Variablen
-        """
-        x, dxds, y, dyds, z, dzds = v
-
-        H = ell.func_H(np.array([x, y, z]))
-        h = dxds**2 + 1/(1-ell.ee**2)*dyds**2 + 1/(1-ell.ex**2)*dzds**2
-
-        ddx = -(h/H)*x
-        ddy = -(h/H)*y/(1-ell.ee**2)
-        ddz = -(h/H)*z/(1-ell.ex**2)
-
-        return np.array([dxds, ddx, dyds, ddy, dzds, ddz])
-    return ODE
-
-def gha1_num(ell: EllipsoidTriaxial, point: NDArray, alpha0: float, s: float, num: int, all_points: bool = False) -> Tuple[NDArray, float] | Tuple[NDArray, float, List]:
-    """
-    Panou, Korakitits 2019
-    :param ell:
-    :param point:
-    :param alpha0:
-    :param s:
-    :param num:
-    :param all_points:
-    :return:
-    """
-    phi, lam, _ = ell.cart2geod(point, "ligas3")
-    p0 = ell.geod2cart(phi, lam, 0)
-    x0, y0, z0 = p0
-
-    p, q = pq_ell(ell, p0)
-    dxds0 = p[0] * sin(alpha0) + q[0] * cos(alpha0)
-    dyds0 = p[1] * sin(alpha0) + q[1] * cos(alpha0)
-    dzds0 = p[2] * sin(alpha0) + q[2] * cos(alpha0)
-
-    v_init = np.array([x0, dxds0, y0, dyds0, z0, dzds0])
-
-    ode = buildODE(ell)
-
-    _, werte = rk.rk4(ode, 0, v_init, s, num)
-    x1, dx1ds, y1, dy1ds, z1, dz1ds = werte[-1]
-
-    point1 = np.array([x1, y1, z1])
-
-    p1, q1 = pq_ell(ell, point1)
-    sigma = np.array([dx1ds, dy1ds, dz1ds])
-    P = float(p1 @ sigma)
-    Q = float(q1 @ sigma)
-
-    alpha1 = arctan2(P, Q)
-
-    if alpha1 < 0:
-        alpha1 += 2 * np.pi
-
-    if all_points:
-        return point1, alpha1, werte
-    else:
-        return point1, alpha1
-
-# ---------------------------------------------------------------------------------------------------------------------
-
-def pq_para(ell: EllipsoidTriaxial, point: NDArray) -> Tuple[NDArray, NDArray]:
-    """
-    Berechnung von p und q in parametrischen Koordinaten
-    Panou, Korakitits 2020
-    :param ell: Ellipsoid
-    :param point: Punkt
-    :return: p und q
-    """
-    n = ell.func_n(point)
-    u, v = ell.cart2para(point)
-
-    # 41-47
-    G = sqrt(1 - ell.ex ** 2 * cos(u) ** 2 - ell.ee ** 2 * sin(u) ** 2 * sin(v) ** 2)
-    q = np.array([-1 / G * sin(u) * cos(v),
-                  -1 / G * sqrt(1 - ell.ee ** 2) * sin(u) * sin(v),
-                  1 / G * sqrt(1 - ell.ex ** 2) * cos(u)])
-    p = np.array([q[1] * n[2] - q[2] * n[1],
-                  q[2] * n[0] - q[0] * n[2],
-                  q[0] * n[1] - q[1] * n[0]])
-
-    t1 = np.dot(n, q)
-    t2 = np.dot(n, p)
-    t3 = np.dot(p, q)
-    if not (t1 < 1e-10 or t1 > 1-1e-10) and not (t2 < 1e-10 or t2 > 1-1e-10) and not (t3 < 1e-10 or t3 > 1-1e-10):
-        raise Exception("Fehler in den normierten Vektoren")
-
-    return p, q
-
-def gha1_ana_step(ell: EllipsoidTriaxial, point: NDArray, alpha0: float, s: float, maxM: int) -> Tuple[NDArray, float]:
-    """
-    Panou, Korakitits 2020, 5ff.
-    :param ell:
-    :param point:
-    :param alpha0:
-    :param s:
-    :param maxM:
-    :return:
-    """
-    x, y, z = point
-
-    # S. 6
-    x_m = [x]
-    y_m = [y]
-    z_m = [z]
-
-    p, q = pq_para(ell, point)
-
-    # 48-50
-    x_m.append(p[0] * sin(alpha0) + q[0] * cos(alpha0))
-    y_m.append(p[1] * sin(alpha0) + q[1] * cos(alpha0))
-    z_m.append(p[2] * sin(alpha0) + q[2] * cos(alpha0))
-
-    # 34
-    H_ = lambda p: np.sum([comb(p, p - i) * (x_m[p - i] * x_m[i] +
-                                             1 / (1 - ell.ee ** 2) ** 2 * y_m[p - i] * y_m[i] +
-                                             1 / (1 - ell.ex ** 2) ** 2 * z_m[p - i] * z_m[i])
-                           for i in range(0, p + 1)])
-
-    # 35
-    h_ = lambda q: np.sum([comb(q, q-j) * (x_m[q-j+1] * x_m[j+1] +
-                                           1 / (1 - ell.ee ** 2) * y_m[q-j+1] * y_m[j+1] +
-                                           1 / (1 - ell.ex ** 2) * z_m[q-j+1] * z_m[j+1])
-                           for j in range(0, q+1)])
-
-    # 31
-    hH_ = lambda t: 1/H_(0) * (h_(t) - np.sum([comb(t, l-1) * H_(t+1-l) * hH_t[l-1] for l in range(1, t+1)]))
-
-    # 28-30
-    x_ = lambda m: - np.sum([comb(m-2, k) * hH_t[m-2-k] * x_m[k] for k in range(0, m-2+1)])
-    y_ = lambda m: -1 / (1-ell.ee**2) * np.sum([comb(m-2, k) * hH_t[m-2-k] * y_m[k] for k in range(0, m-2+1)])
-    z_ = lambda m: -1 / (1-ell.ex**2) * np.sum([comb(m-2, k) * hH_t[m-2-k] * z_m[k] for k in range(0, m-2+1)])
-
-    hH_t = []
-    a_m = []
-    b_m = []
-    c_m = []
-    for m in range(0, maxM+1):
-        if m >= 2:
-            hH_t.append(hH_(m-2))
-            x_m.append(x_(m))
-            y_m.append(y_(m))
-            z_m.append(z_(m))
-        fact_m = fact(m)
-
-        # 22-24
-        a_m.append(x_m[m] / fact_m)
-        b_m.append(y_m[m] / fact_m)
-        c_m.append(z_m[m] / fact_m)
-
-    # 19-21
-    x_s = 0
-    for a in reversed(a_m):
-        x_s = x_s * s + a
-    y_s = 0
-    for b in reversed(b_m):
-        y_s = y_s * s + b
-    z_s = 0
-    for c in reversed(c_m):
-        z_s = z_s * s + c
-
-    p1 = np.array([x_s, y_s, z_s])
-    p_s, q_s = pq_para(ell, p1)
-
-    # 57-59
-    dx_s = 0
-    for i, a in reversed(list(enumerate(a_m[1:], start=1))):
-        dx_s = dx_s * s + i * a
-
-    dy_s = 0
-    for i, b in reversed(list(enumerate(b_m[1:], start=1))):
-        dy_s = dy_s * s + i * b
-
-    dz_s = 0
-    for i, c in reversed(list(enumerate(c_m[1:], start=1))):
-        dz_s = dz_s * s + i * c
-
-    # 52-53
-    sigma = np.array([dx_s, dy_s, dz_s])
-    P = float(p_s @ sigma)
-    Q = float(q_s @ sigma)
-
-    # 51
-    alpha1 = arctan2(P, Q)
-
-    if alpha1 < 0:
-        alpha1 += 2 * np.pi
-
-    return p1, alpha1
-
-
-def gha1_ana(ell: EllipsoidTriaxial, point: NDArray, alpha0: float, s: float, maxM: int, maxPartCircum: int = 4) -> Tuple[NDArray, float]:
-    if s > np.pi / maxPartCircum * ell.ax:
-        s /= 2
-        point_step, alpha_step = gha1_ana(ell, point, alpha0, s, maxM, maxPartCircum)
-        point_end, alpha_end = gha1_ana(ell, point_step, alpha_step, s, maxM, maxPartCircum)
-    else:
-        point_end, alpha_end = gha1_ana_step(ell, point, alpha0, s, maxM)
-
-    _, _, h = ell.cart2geod(point_end, "ligas3")
-    if h > 1e-5:
-        raise Exception("Analyitsche Methode ist explodiert, Punkt liegt nicht mehr auf dem Ellpsoid")
-
-    return point_end, alpha_end
-
-def alpha_para2ell(ell: EllipsoidTriaxial, u: float, v: float, alpha_para: float) -> Tuple[float, float, float]:
-    point = ell.para2cart(u, v)
-    beta, lamb = ell.para2ell(u, v)
-
-    p_para, q_para = pq_para(ell, point)
-    sigma_para = p_para * sin(alpha_para) + q_para * cos(alpha_para)
-
-    p_ell, q_ell = pq_ell(ell, point)
-    alpha_ell = arctan2(p_ell @ sigma_para, q_ell @ sigma_para)
-    sigma_ell = p_ell * sin(alpha_ell) + q_ell * cos(alpha_ell)
-
-    if np.linalg.norm(sigma_para - sigma_ell) > 1e-12:
-        raise Exception("Alpha Umrechnung fehlgeschlagen")
-
-    return beta, lamb, alpha_ell
-
-def alpha_ell2para(ell: EllipsoidTriaxial, beta: float, lamb: float, alpha_ell: float) -> Tuple[float, float, float]:
-    point = ell.ell2cart(beta, lamb)
-    u, v = ell.ell2para(beta, lamb)
-
-    p_ell, q_ell = pq_ell(ell, point)
-    sigma_ell = p_ell * sin(alpha_ell) + q_ell * cos(alpha_ell)
-
-    p_para, q_para = pq_para(ell, point)
-    alpha_para = arctan2(p_para @ sigma_ell, q_para @ sigma_ell)
-    sigma_para = p_para * sin(alpha_para) + q_para * cos(alpha_para)
-
-    if np.linalg.norm(sigma_para - sigma_ell) > 1e-12:
-        raise Exception("Alpha Umrechnung fehlgeschlagen")
-
-    return u, v, alpha_para
-
-def func_sigma_ell(ell: EllipsoidTriaxial, point: NDArray, alpha: float) -> NDArray:
-    p, q = pq_ell(ell, point)
-    sigma = p * sin(alpha) + q * cos(alpha)
-    return sigma
-
-def func_sigma_para(ell: EllipsoidTriaxial, point: NDArray, alpha: float) -> NDArray:
-    p, q = pq_para(ell, point)
-    sigma = p * sin(alpha) + q * cos(alpha)
-    return sigma
-
-
-def louville_constant(ell: EllipsoidTriaxial, p0: NDArray, alpha: float) -> float:
-    beta, lamb = ell.cart2ell(p0)
-    l = ell.Ey**2 * cos(beta)**2 * sin(alpha)**2 - ell.Ee**2 * sin(lamb)**2 * cos(alpha)**2
-    return l
-
-def louville_l2c(ell: EllipsoidTriaxial, l: float) -> float:
-    return sqrt((l + ell.Ee**2) / ell.Ex**2)
-
-def louville_c2l(ell: EllipsoidTriaxial, c: float) -> float:
-    return ell.Ex**2 * c**2 - ell.Ee**2
-
-
-if __name__ == "__main__":
-
-    # ell = ellipsoide.EllipsoidTriaxial.init_name("BursaSima1980round")
-    # diffs_panou = []
-    # examples_panou = ne_panou.get_random_examples(5)
-    # for example in examples_panou:
-    #     beta0, lamb0, beta1, lamb1, _, alpha0_ell, alpha1_ell, s = example
-    #     P0 = ell.ell2cart(beta0, lamb0)
-    #
-    #     P1_num, alpha1_num = gha1_num(ell, P0, alpha0_ell, s, 100)
-    #     beta1_num, lamb1_num = ell.cart2ell(P1_num)
-    #
-    #     _, _, alpha0_para = alpha_ell2para(ell, beta0, lamb0, alpha0_ell)
-    #     P1_ana, alpha1_ana = gha1_ana(ell, P0, alpha0_para, s, 60)
-    #     beta1_ana, lamb1_ana = ell.cart2ell(P1_ana)
-    #     diffs_panou.append((abs(beta1-beta1_num), abs(lamb1-lamb1_num), abs(beta1-beta1_ana), abs(lamb1-lamb1_ana)))
-    # diffs_panou = np.array(diffs_panou)
-    # mask_360 = (diffs_panou > 359) & (diffs_panou < 361)
-    # diffs_panou[mask_360] = np.abs(diffs_panou[mask_360] - 360)
-    # print(diffs_panou)
-
-    ell = ellipsoide.EllipsoidTriaxial.init_name("KarneyTest2024")
-    diffs_karney = []
-    # examples_karney = ne_karney.get_examples((30499, 30500, 40500))
-    examples_karney = ne_karney.get_random_examples(20)
-    for example in examples_karney:
-        beta0, lamb0, alpha0_ell, beta1, lamb1, alpha1_ell, s = example
-        P0 = ell.ell2cart(beta0, lamb0)
-
-        P1_num, alpha1_num = gha1_num(ell, P0, alpha0_ell, s, 5000)
-        beta1_num, lamb1_num = ell.cart2ell(P1_num)
-
-        try:
-            _, _, alpha0_para = alpha_ell2para(ell, beta0, lamb0, alpha0_ell)
-            P1_ana, alpha1_ana = gha1_ana(ell, P0, alpha0_para, s, 30, maxPartCircum=16)
-            beta1_ana, lamb1_ana = ell.cart2ell(P1_ana)
-        except:
-            beta1_ana, lamb1_ana = np.inf, np.inf
-
-        diffs_karney.append((wu.rad2deg(abs(beta1-beta1_num)), wu.rad2deg(abs(lamb1-lamb1_num)), wu.rad2deg(abs(beta1-beta1_ana)), wu.rad2deg(abs(lamb1-lamb1_ana))))
-    diffs_karney = np.array(diffs_karney)
-    mask_360 = (diffs_karney > 359) & (diffs_karney < 361)
-    diffs_karney[mask_360] = np.abs(diffs_karney[mask_360] - 360)
-    print(diffs_karney)

GHA_triaxial/panou_2013_2GHA_num.py

@@ -1,435 +0,0 @@
-import numpy as np
-from ellipsoide import EllipsoidTriaxial
-import runge_kutta as rk
-import GHA_triaxial.numeric_examples_karney as ne_karney
-import GHA_triaxial.numeric_examples_panou as ne_panou
-import winkelumrechnungen as wu
-from typing import Tuple
-from numpy.typing import NDArray
-
-# Panou 2013
-def gha2_num(ell: EllipsoidTriaxial, beta_1: float, lamb_1: float, beta_2: float, lamb_2: float,
-             n: int = 16000, epsilon: float = 10**-12, iter_max: int = 30, all_points: bool = False
-             ) -> Tuple[float, float, float] | Tuple[float, float, float, NDArray, NDArray]:
-    """
-
-    :param ell: triaxiales Ellipsoid
-    :param beta_1: reduzierte ellipsoidische Breite Punkt 1
-    :param lamb_1: elllipsoidische Länge Punkt 1
-    :param beta_2: reduzierte ellipsoidische Breite Punkt 2
-    :param lamb_2: elllipsoidische Länge Punkt 2
-    :param n: Anzahl Schritte
-    :param epsilon:
-    :param iter_max: Maximale Anzhal Iterationen
-    :param all_points:
-    :return:
-    """
-
-    # h_x, h_y, h_e entsprechen E_x, E_y, E_e
-
-    def arccot(x):
-        return np.arctan2(1.0, x)
-
-    def BETA_LAMBDA(beta, lamb):
-
-        BETA = (ell.ay**2 * np.sin(beta)**2 + ell.b**2 * np.cos(beta)**2) / (ell.Ex**2 - ell.Ey**2 * np.sin(beta)**2)
-        LAMBDA = (ell.ax**2 * np.sin(lamb)**2 + ell.ay**2 * np.cos(lamb)**2) / (ell.Ex**2 - ell.Ee**2 * np.cos(lamb)**2)
-
-        # Erste Ableitungen von ΒETA und LAMBDA
-        BETA_ = (ell.ax**2 * ell.Ey**2 * np.sin(2*beta)) / (ell.Ex**2 - ell.Ey**2 * np.sin(beta)**2)**2
-        LAMBDA_ = - (ell.b**2 * ell.Ee**2 * np.sin(2*lamb)) / (ell.Ex**2 - ell.Ee**2 * np.cos(lamb)**2)**2
-
-        # Zweite Ableitungen von ΒETA und LAMBDA
-        BETA__ = ((2 * ell.ax**2 * ell.Ey**4 * np.sin(2*beta)**2) / (ell.Ex**2 - ell.Ey**2 * np.sin(beta)**2)**3) + ((2 * ell.ax**2 * ell.Ey**2 * np.cos(2*beta)) / (ell.Ex**2 - ell.Ey**2 * np.sin(beta)**2)**2)
-        LAMBDA__ = (((2 * ell.b**2 * ell.Ee**4 * np.sin(2*lamb)**2) / (ell.Ex**2 - ell.Ee**2 * np.cos(lamb)**2)**3) -
-                    ((2 * ell.b**2 * ell.Ee**2 * np.sin(2*lamb)) / (ell.Ex**2 - ell.Ee**2 * np.cos(lamb)**2)**2))
-
-        E = BETA * (ell.Ey ** 2 * np.cos(beta) ** 2 + ell.Ee ** 2 * np.sin(lamb) ** 2)
-        F = 0
-        G = LAMBDA * (ell.Ey ** 2 * np.cos(beta) ** 2 + ell.Ee ** 2 * np.sin(lamb) ** 2)
-
-        # Erste Ableitungen von E und G
-        E_beta = BETA_ * (ell.Ey**2 * np.cos(beta)**2 + ell.Ee**2 * np.sin(lamb)**2) - BETA * ell.Ey**2 * np.sin(2*beta)
-        E_lamb = BETA * ell.Ee**2 * np.sin(2*lamb)
-
-        G_beta = - LAMBDA * ell.Ey**2 * np.sin(2*beta)
-        G_lamb = LAMBDA_ * (ell.Ey**2 * np.cos(beta)**2 + ell.Ee**2 * np.sin(lamb)**2) + LAMBDA * ell.Ee**2 * np.sin(2*lamb)
-
-        # Zweite Ableitungen von E und G
-        E_beta_beta = BETA__ * (ell.Ey**2 * np.cos(beta)**2 + ell.Ee**2 * np.sin(lamb)**2) - 2 * BETA_ * ell.Ey**2 * np.sin(2*beta) - 2 * BETA * ell.Ey**2 * np.cos(2*beta)
-        E_beta_lamb = BETA_ * ell.Ee**2 * np.sin(2*lamb)
-        E_lamb_lamb = 2 * BETA * ell.Ee**2 * np.cos(2*lamb)
-
-        G_beta_beta = - 2 * LAMBDA * ell.Ey**2 * np.cos(2*beta)
-        G_beta_lamb = - LAMBDA_ * ell.Ey**2 * np.sin(2*beta)
-        G_lamb_lamb = LAMBDA__ * (ell.Ey**2 * np.cos(beta)**2 + ell.Ee**2 * np.sin(lamb)**2) + 2 * LAMBDA_ * ell.Ee**2 * np.sin(2*lamb) + 2 * LAMBDA * ell.Ee**2 * np.cos(2*lamb)
-
-        return (BETA, LAMBDA, E, G,
-                BETA_, LAMBDA_, BETA__, LAMBDA__,
-                E_beta, E_lamb, G_beta, G_lamb,
-                E_beta_beta, E_beta_lamb, E_lamb_lamb,
-                G_beta_beta, G_beta_lamb, G_lamb_lamb)
-
-    def p_coef(beta, lamb):
-
-        (BETA, LAMBDA, E, G,
-         BETA_, LAMBDA_, BETA__, LAMBDA__,
-         E_beta, E_lamb, G_beta, G_lamb,
-         E_beta_beta, E_beta_lamb, E_lamb_lamb,
-         G_beta_beta, G_beta_lamb, G_lamb_lamb) = BETA_LAMBDA(beta, lamb)
-
-        p_3 = - 0.5 * (E_lamb / G)
-        p_2 = (G_beta / G) - 0.5 * (E_beta / E)
-        p_1 = 0.5 * (G_lamb / G) - (E_lamb / E)
-        p_0 = 0.5 * (G_beta / E)
-
-        p_33 = - 0.5 * ((E_beta_lamb * G - E_lamb * G_beta) / (G ** 2))
-        p_22 = ((G * G_beta_beta - G_beta * G_beta) / (G ** 2)) - 0.5 * ((E * E_beta_beta - E_beta * E_beta) / (E ** 2))
-        p_11 = 0.5 * ((G * G_beta_lamb - G_beta * G_lamb) / (G ** 2)) - ((E * E_beta_lamb - E_beta * E_lamb) / (E ** 2))
-        p_00 = 0.5 * ((E * G_beta_beta - E_beta * G_beta) / (E ** 2))
-
-        return (BETA, LAMBDA, E, G,
-                p_3, p_2, p_1, p_0,
-                p_33, p_22, p_11, p_00)
-
-    def q_coef(beta, lamb):
-
-        (BETA, LAMBDA, E, G,
-         BETA_, LAMBDA_, BETA__, LAMBDA__,
-         E_beta, E_lamb, G_beta, G_lamb,
-         E_beta_beta, E_beta_lamb, E_lamb_lamb,
-         G_beta_beta, G_beta_lamb, G_lamb_lamb) = BETA_LAMBDA(beta, lamb)
-
-        q_3 = - 0.5 * (G_beta / E)
-        q_2 = (E_lamb / E) - 0.5 * (G_lamb / G)
-        q_1 = 0.5 * (E_beta / E) - (G_beta / G)
-        q_0 = 0.5 * (E_lamb / G)
-
-        q_33 = - 0.5 * ((E * G_beta_lamb - E_lamb * G_lamb) / (E ** 2))
-        q_22 = ((E * E_lamb_lamb - E_lamb * E_lamb) / (E ** 2)) - 0.5 * ((G * G_lamb_lamb - G_lamb * G_lamb) / (G ** 2))
-        q_11 = 0.5 * ((E * E_beta_lamb - E_beta * E_lamb) / (E ** 2)) - ((G * G_beta_lamb - G_beta * G_lamb) / (G ** 2))
-        q_00 = 0.5 * ((E_lamb_lamb * G - E_lamb * G_lamb) / (G ** 2))
-
-        return (BETA, LAMBDA, E, G,
-                q_3, q_2, q_1, q_0,
-                q_33, q_22, q_11, q_00)
-
-    if lamb_1 != lamb_2:
-        # def functions():
-        #     def f_beta(lamb, beta, beta_p, X3, X4):
-        #         return beta_p
-        #
-        #     def f_beta_p(lamb, beta, beta_p, X3, X4):
-        #         (BETA, LAMBDA, E, G,
-        #          p_3, p_2, p_1, p_0,
-        #          p_33, p_22, p_11, p_00) = p_coef(beta, lamb)
-        #         return p_3 * beta_p ** 3 + p_2 * beta_p ** 2 + p_1 * beta_p + p_0
-        #
-        #     def f_X3(lamb, beta, beta_p, X3, X4):
-        #         return X4
-        #
-        #     def f_X4(lamb, beta, beta_p, X3, X4):
-        #         (BETA, LAMBDA, E, G,
-        #          p_3, p_2, p_1, p_0,
-        #          p_33, p_22, p_11, p_00) = p_coef(beta, lamb)
-        #         return (p_33 * beta_p ** 3 + p_22 * beta_p ** 2 + p_11 * beta_p + p_00) * X3 + \
-        #             (3 * p_3 * beta_p ** 2 + 2 * p_2 * beta_p + p_1) * X4
-        #
-        #     return [f_beta, f_beta_p, f_X3, f_X4]
-
-        def buildODElamb():
-            def ODE(lamb, v):
-                beta, beta_p, X3, X4 = v
-
-                (BETA, LAMBDA, E, G,
-                 p_3, p_2, p_1, p_0,
-                 p_33, p_22, p_11, p_00) = p_coef(beta, lamb)
-
-                dbeta = beta_p
-                dbeta_p = p_3 * beta_p ** 3 + p_2 * beta_p ** 2 + p_1 * beta_p + p_0
-                dX3 = X4
-                dX4 = (p_33 * beta_p ** 3 + p_22 * beta_p ** 2 + p_11 * beta_p + p_00) * X3 + \
-                    (3 * p_3 * beta_p ** 2 + 2 * p_2 * beta_p + p_1) * X4
-
-                return np.array([dbeta, dbeta_p, dX3, dX4])
-
-            return ODE
-
-        N = n
-
-        dlamb = lamb_2 - lamb_1
-
-        if abs(dlamb) < 1e-15:
-            beta_0 = 0.0
-        else:
-            beta_0 = (beta_2 - beta_1) / (lamb_2 - lamb_1)
-
-        converged = False
-        iterations = 0
-
-        # funcs = functions()
-        ode_lamb = buildODElamb()
-
-        for i in range(iter_max):
-            iterations = i + 1
-
-            # startwerte = [lamb_1, beta_1, beta_0, 0.0, 1.0]
-            startwerte = np.array([beta_1, beta_0, 0.0, 1.0])
-
-            # werte = rk.verfahren(funcs, startwerte, dlamb, N)
-            lamb_list, werte = rk.rk4(ode_lamb, lamb_1, startwerte, dlamb, N, False)
-            # lamb_end, beta_end, beta_p_end, X3_end, X4_end = werte[-1]
-            lamb_end = lamb_list[-1]
-            beta_end, beta_p_end, X3_end, X4_end = werte[-1]
-
-            d_beta_end_d_beta0 = X3_end
-            delta = beta_end - beta_2
-
-            if abs(delta) < epsilon:
-                converged = True
-                break
-
-            if abs(d_beta_end_d_beta0) < 1e-20:
-                raise RuntimeError("Abbruch.")
-
-            max_step = 0.5
-            step = delta / d_beta_end_d_beta0
-            if abs(step) > max_step:
-                step = np.sign(step) * max_step
-            beta_0 = beta_0 - step
-
-        if not converged:
-            raise RuntimeError("konvergiert nicht.")
-
-        # Z
-        # werte = rk.verfahren(funcs, [lamb_1, beta_1, beta_0, 0.0, 1.0], dlamb, N, False)
-        lamb_list, werte = rk.rk4(ode_lamb, lamb_1, np.array([beta_1, beta_0, 0.0, 1.0]), dlamb, N, False)
-
-        beta_arr = np.zeros(N + 1)
-        # lamb_arr = np.zeros(N + 1)
-        lamb_arr = np.array(lamb_list)
-        beta_p_arr = np.zeros(N + 1)
-
-        for i, state in enumerate(werte):
-            # lamb_arr[i] = state[0]
-            # beta_arr[i] = state[1]
-            # beta_p_arr[i] = state[2]
-            beta_arr[i] = state[0]
-            beta_p_arr[i] = state[1]
-
-        (_, _, E1, G1,
-         *_) = BETA_LAMBDA(beta_arr[0], lamb_arr[0])
-        (_, _, E2, G2,
-         *_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
-
-        alpha_1 = arccot(np.sqrt(E1 / G1) * beta_p_arr[0])
-        alpha_2 = arccot(np.sqrt(E2 / G2) * beta_p_arr[-1])
-
-        integrand = np.zeros(N + 1)
-        for i in range(N + 1):
-            (_, _, Ei, Gi,
-             *_) = BETA_LAMBDA(beta_arr[i], lamb_arr[i])
-            integrand[i] = np.sqrt(Ei * beta_p_arr[i] ** 2 + Gi)
-
-        h = abs(dlamb) / N
-        if N % 2 == 0:
-            S = integrand[0] + integrand[-1] \
-                + 4.0 * np.sum(integrand[1:-1:2]) \
-                + 2.0 * np.sum(integrand[2:-1:2])
-            s = h / 3.0 * S
-        else:
-            s = np.trapz(integrand, dx=h)
-
-        beta0 = beta_arr[0]
-        lamb0 = lamb_arr[0]
-        c = np.sqrt(
-            (np.cos(beta0) ** 2 + (ell.Ee**2 / ell.Ex**2) * np.sin(beta0) ** 2) * np.sin(alpha_1) ** 2
-            + (ell.Ee**2 / ell.Ex**2) * np.cos(lamb0) ** 2 * np.cos(alpha_1) ** 2
-        )
-
-        if all_points:
-            return alpha_1, alpha_2, s, beta_arr, lamb_arr
-        else:
-            return alpha_1, alpha_2, s
-
-    else:  # lamb_1 == lamb_2
-
-        N = n
-        dbeta = beta_2 - beta_1
-
-        if abs(dbeta) < 10**-15:
-            if all_points:
-                return 0, 0, 0, np.array([]), np.array([])
-            else:
-                return 0, 0, 0
-
-        lamb_0 = 0
-
-        converged = False
-        iterations = 0
-
-        # def functions_beta():
-        #     def g_lamb(beta, lamb, lamb_p, Y3, Y4):
-        #         return lamb_p
-        #
-        #     def g_lamb_p(beta, lamb, lamb_p, Y3, Y4):
-        #         (BETA, LAMBDA, E, G,
-        #          q_3, q_2, q_1, q_0,
-        #          q_33, q_22, q_11, q_00) = q_coef(beta, lamb)
-        #         return q_3 * lamb_p ** 3 + q_2 * lamb_p ** 2 + q_1 * lamb_p + q_0
-        #
-        #     def g_Y3(beta, lamb, lamb_p, Y3, Y4):
-        #         return Y4
-        #
-        #     def g_Y4(beta, lamb, lamb_p, Y3, Y4):
-        #         (BETA, LAMBDA, E, G,
-        #          q_3, q_2, q_1, q_0,
-        #          q_33, q_22, q_11, q_00) = q_coef(beta, lamb)
-        #         return (q_33 * lamb_p ** 3 + q_22 * lamb_p ** 2 + q_11 * lamb_p + q_00) * Y3 + \
-        #             (3 * q_3 * lamb_p ** 2 + 2 * q_2 * lamb_p + q_1) * Y4
-        #
-        #     return [g_lamb, g_lamb_p, g_Y3, g_Y4]
-
-        def buildODEbeta():
-            def ODE(beta, v):
-                lamb, lamb_p, Y3, Y4 = v
-
-                (BETA, LAMBDA, E, G,
-                 q_3, q_2, q_1, q_0,
-                 q_33, q_22, q_11, q_00) = q_coef(beta, lamb)
-
-                dlamb = lamb_p
-                dlamb_p = q_3 * lamb_p ** 3 + q_2 * lamb_p ** 2 + q_1 * lamb_p + q_0
-                dY3 = Y4
-                dY4 = (q_33 * lamb_p ** 3 + q_22 * lamb_p ** 2 + q_11 * lamb_p + q_00) * Y3 + \
-                    (3 * q_3 * lamb_p ** 2 + 2 * q_2 * lamb_p + q_1) * Y4
-
-                return np.array([dlamb, dlamb_p, dY3, dY4])
-            return ODE
-
-        # funcs_beta = functions_beta()
-        ode_beta = buildODEbeta()
-
-        for i in range(iter_max):
-            iterations = i + 1
-
-            startwerte = [lamb_1, lamb_0, 0.0, 1.0]
-
-            # werte = rk.verfahren(funcs_beta, startwerte, dbeta, N, False)
-            beta_list, werte = rk.rk4(ode_beta, beta_1, startwerte, dbeta, N, False)
-
-            beta_end = beta_list[-1]
-            # beta_end, lamb_end, lamb_p_end, Y3_end, Y4_end = werte[-1]
-            lamb_end, lamb_p_end, Y3_end, Y4_end = werte[-1]
-
-            d_lamb_end_d_lambda0 = Y3_end
-            delta = lamb_end - lamb_2
-
-            if abs(delta) < epsilon:
-                converged = True
-                break
-
-            if abs(d_lamb_end_d_lambda0) < 1e-20:
-                raise RuntimeError("Abbruch (Ableitung ~ 0).")
-
-            max_step = 1.0
-            step = delta / d_lamb_end_d_lambda0
-            if abs(step) > max_step:
-                step = np.sign(step) * max_step
-
-            lamb_0 = lamb_0 - step
-
-        # werte = rk.verfahren(funcs_beta, [beta_1, lamb_1, lamb_0, 0.0, 1.0], dbeta, N, False)
-        beta_list, werte = rk.rk4(ode_beta, beta_1, np.array([lamb_1, lamb_0, 0.0, 1.0]), dbeta, N, False)
-
-        # beta_arr = np.zeros(N + 1)
-        beta_arr = np.array(beta_list)
-        lamb_arr = np.zeros(N + 1)
-        lambda_p_arr = np.zeros(N + 1)
-
-        for i, state in enumerate(werte):
-            # beta_arr[i] = state[0]
-            # lamb_arr[i] = state[1]
-            # lambda_p_arr[i] = state[2]
-            lamb_arr[i] = state[0]
-            lambda_p_arr[i] = state[1]
-
-        # Azimute
-        (BETA1, LAMBDA1, E1, G1,
-         *_) = BETA_LAMBDA(beta_arr[0], lamb_arr[0])
-        (BETA2, LAMBDA2, E2, G2,
-         *_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
-
-        alpha_1 = (np.pi / 2.0) - arccot(np.sqrt(LAMBDA1 / BETA1) * lambda_p_arr[0])
-        alpha_2 = (np.pi / 2.0) - arccot(np.sqrt(LAMBDA2 / BETA2) * lambda_p_arr[-1])
-
-        integrand = np.zeros(N + 1)
-        for i in range(N + 1):
-            (_, _, Ei, Gi,
-             *_) = BETA_LAMBDA(beta_arr[i], lamb_arr[i])
-            integrand[i] = np.sqrt(Ei + Gi * lambda_p_arr[i] ** 2)
-
-        h = abs(dbeta) / N
-        if N % 2 == 0:
-            S = integrand[0] + integrand[-1] \
-                + 4.0 * np.sum(integrand[1:-1:2]) \
-                + 2.0 * np.sum(integrand[2:-1:2])
-            s = h / 3.0 * S
-        else:
-            s = np.trapz(integrand, dx=h)
-
-        if all_points:
-            return alpha_1, alpha_2, s, beta_arr, lamb_arr
-        else:
-            return alpha_1, alpha_2, s
-
-
-if __name__ == "__main__":
-    # ell = EllipsoidTriaxial.init_name("Fiction")
-    # # beta1 = np.deg2rad(75)
-    # # lamb1 = np.deg2rad(-90)
-    # # beta2 = np.deg2rad(75)
-    # # lamb2 = np.deg2rad(66)
-    # # a1, a2, s = gha2_num(ell, beta1, lamb1, beta2, lamb2)
-    # # print(aus.gms("a1", a1, 4))
-    # # print(aus.gms("a2", a2, 4))
-    # # print(s)
-    # cart1 = ell.para2cart(0, 0)
-    # cart2 = ell.para2cart(0.4, 1.4)
-    # beta1, lamb1 = ell.cart2ell(cart1)
-    # beta2, lamb2 = ell.cart2ell(cart2)
-    #
-    # a1, a2, s = gha2_num(ell, beta1, lamb1, beta2, lamb2, n=5000)
-    # print(s)
-
-    # ell = EllipsoidTriaxial.init_name("BursaSima1980round")
-    # diffs_panou = []
-    # examples_panou = ne_panou.get_random_examples(4)
-    # for example in examples_panou:
-    #     beta0, lamb0, beta1, lamb1, _, alpha0, alpha1, s = example
-    #     P0 = ell.ell2cart(beta0, lamb0)
-    #     try:
-    #         alpha0_num, alpha1_num, s_num = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=4000, iter_max=10)
-    #         diffs_panou.append(
-    #             (wu.rad2deg(abs(alpha0 - alpha0_num)), wu.rad2deg(abs(alpha1 - alpha1_num)), abs(s - s_num)))
-    #     except:
-    #         print(f"Fehler für {beta0}, {lamb0}, {beta1}, {lamb1}")
-    # diffs_panou = np.array(diffs_panou)
-    # print(diffs_panou)
-    #
-    # ell = EllipsoidTriaxial.init_name("KarneyTest2024")
-    # diffs_karney = []
-    # # examples_karney = ne_karney.get_examples((30500, 40500))
-    # examples_karney = ne_karney.get_random_examples(2)
-    # for example in examples_karney:
-    #     beta0, lamb0, alpha0, beta1, lamb1, alpha1, s = example
-    #
-    #     try:
-    #         alpha0_num, alpha1_num, s_num = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=4000, iter_max=10)
-    #         diffs_karney.append((wu.rad2deg(abs(alpha0-alpha0_num)), wu.rad2deg(abs(alpha1-alpha1_num)), abs(s-s_num)))
-    #     except:
-    #         print(f"Fehler für {beta0}, {lamb0}, {beta1}, {lamb1}")
-    # diffs_karney = np.array(diffs_karney)
-    # print(diffs_karney)
-
-    pass

GHA_triaxial/utils.py

@@ -0,0 +1,201 @@
+from __future__ import annotations
+
+from typing import Tuple
+
+import numpy as np
+from numpy import arctan2, cos, sin, sqrt
+from numpy.typing import NDArray
+
+from ellipsoid_triaxial import EllipsoidTriaxial
+from utils_angle import wrap_0_2pi
+
+
+def sigma2alpha(ell: EllipsoidTriaxial, sigma: NDArray, point: NDArray) -> float:
+    """
+    Berechnung des Azimuts an einem Punkt anhand der Ableitung zu den kartesischen Koordinaten
+    :param ell: Ellipsoid
+    :param sigma: Ableitungsvektor ver kartesischen Koordinaten
+    :param point: Punkt
+    :return: Azimuts
+    """
+    p, q = pq_ell(ell, point)
+    P = float(p @ sigma)
+    Q = float(q @ sigma)
+
+    alpha = arctan2(P, Q)
+    return wrap_0_2pi(alpha)
+
+
+def alpha_para2ell(ell: EllipsoidTriaxial, u: float, v: float, alpha_para: float) -> Tuple[float, float, float]:
+    """
+    Umrechnung des Azimuts bezogen auf parametrische Koordinaten zu ellipsoidischen
+    :param ell: Ellipsoid
+    :param u: parametrische Breite
+    :param v: parametrische Länge
+    :param alpha_para: Azimut bezogen auf parametrische Koordinaten
+    :return: Azimut bezogen auf ellipsoidische Koordinaten
+    """
+    point = ell.para2cart(u, v)
+    beta, lamb = ell.para2ell(u, v)
+
+    p_para, q_para = pq_para(ell, point)
+    sigma_para = p_para * sin(alpha_para) + q_para * cos(alpha_para)
+
+    p_ell, q_ell = pq_ell(ell, point)
+    alpha_ell = arctan2(p_ell @ sigma_para, q_ell @ sigma_para)
+    sigma_ell = p_ell * sin(alpha_ell) + q_ell * cos(alpha_ell)
+
+    if np.linalg.norm(sigma_para - sigma_ell) > 1e-7:
+        raise Exception("alpha_para2ell: Differenz in den Richtungsableitungen")
+
+    return beta, lamb, wrap_0_2pi(alpha_ell)
+
+
+def alpha_ell2para(ell: EllipsoidTriaxial, beta: float, lamb: float, alpha_ell: float) -> Tuple[float, float, float]:
+    """
+    Umrechnung des Azimuts bezogen auf ellipsoidische Koordinaten zu parametrischen
+    :param ell: Ellipsoid
+    :param beta: ellipsoidische Breite
+    :param lamb: ellipsoidische Länge
+    :param alpha_ell: Azimut bezogen auf ellipsoidische Koordinaten
+    :return: Azimut bezogen auf parametrische Koordinaten
+    """
+    point = ell.ell2cart(beta, lamb)
+    u, v = ell.ell2para(beta, lamb)
+
+    p_ell, q_ell = pq_ell(ell, point)
+    sigma_ell = p_ell * sin(alpha_ell) + q_ell * cos(alpha_ell)
+
+    p_para, q_para = pq_para(ell, point)
+    alpha_para = arctan2(p_para @ sigma_ell, q_para @ sigma_ell)
+    sigma_para = p_para * sin(alpha_para) + q_para * cos(alpha_para)
+
+    if np.linalg.norm(sigma_para - sigma_ell) > 1e-7:
+        raise Exception("alpha_ell2para: Differenz in den Richtungsableitungen")
+
+    return u, v, wrap_0_2pi(alpha_para)
+
+
+def func_sigma_ell(ell: EllipsoidTriaxial, point: NDArray, alpha_ell: float) -> NDArray:
+    """
+    Berechnung der Richtungsableitungen in kartesischen Koordinaten aus ellipsoidischem Azimut
+    Panou (2019) [6]
+    :param ell: Ellipsoid
+    :param point: Punkt in kartesischen Koordinaten
+    :param alpha_ell: ellipsoidischer Azimut
+    :return: Richtungsableitungen in kartesischen Koordinaten
+    """
+    p, q = pq_ell(ell, point)
+    sigma = p * sin(alpha_ell) + q * cos(alpha_ell)
+    return sigma
+
+
+def func_sigma_para(ell: EllipsoidTriaxial, point: NDArray, alpha_para: float) -> NDArray:
+    """
+    Berechnung der Richtungsableitungen in kartesischen Koordinaten aus parametischem Azimut
+    Panou, Korakitis (2019) [6]
+    :param ell: Ellipsoid
+    :param point: Punkt in kartesischen Koordinaten
+    :param alpha_para: parametrischer Azimut
+    :return: Richtungsableitungen in kartesischen Koordinaten
+    """
+    p, q = pq_para(ell, point)
+    sigma = p * sin(alpha_para) + q * cos(alpha_para)
+    return sigma
+
+
+def louville_constant(ell: EllipsoidTriaxial, p0: NDArray, alpha_ell: float) -> float:
+    """
+    Berechnung der Louville Konstanten
+    Panou, Korakitis (2019) [6]
+    :param ell: Ellipsoid
+    :param p0: Punkt in kartesischen Koordinaten
+    :param alpha_ell: ellipsoidischer Azimut
+    :return:
+    """
+    beta, lamb = ell.cart2ell(p0)
+    l = ell.Ey**2 * cos(beta)**2 * sin(alpha_ell)**2 - ell.Ee**2 * sin(lamb)**2 * cos(alpha_ell)**2
+    return l
+
+
+def pq_ell(ell: EllipsoidTriaxial, point: NDArray) -> Tuple[NDArray, NDArray]:
+    """
+    Berechnung von p (Tangente entlang konstantem beta) und q (Tangente entlang konstantem lambda)
+    Panou, Korakitits (2019) [5f.]
+    :param ell: Ellipsoid
+    :param point: Punkt
+    :return: p und q
+    """
+    n = ell.func_n(point)
+
+    beta, lamb = ell.cart2ell(point)
+    if abs(cos(beta)) < 1e-15 and abs(np.sin(lamb)) < 1e-15:
+        if beta > 0:
+            p = np.array([0, -1, 0])
+        else:
+            p = np.array([0, 1, 0])
+    else:
+        B = ell.Ex ** 2 * cos(beta) ** 2 + ell.Ee ** 2 * sin(beta) ** 2
+        L = ell.Ex ** 2 - ell.Ee ** 2 * cos(lamb) ** 2
+
+        _, t2 = ell.func_t12(point)
+
+        F = ell.Ey ** 2 * cos(beta) ** 2 + ell.Ee ** 2 * sin(lamb) ** 2
+        p1 = -sqrt(L / (F * t2)) * ell.ax / ell.Ex * sqrt(B) * sin(lamb)
+        p2 = sqrt(L / (F * t2)) * ell.ay * cos(beta) * cos(lamb)
+        p3 = 1 / sqrt(F * t2) * (ell.b * ell.Ee ** 2) / (2 * ell.Ex) * sin(beta) * sin(2 * lamb)
+        p = np.array([p1, p2, p3])
+        p = p / np.linalg.norm(p)
+    q = np.array([n[1] * p[2] - n[2] * p[1],
+                  n[2] * p[0] - n[0] * p[2],
+                  n[0] * p[1] - n[1] * p[0]])
+    q = q / np.linalg.norm(q)
+
+    return p, q
+
+
+def pq_para(ell: EllipsoidTriaxial, point: NDArray) -> Tuple[NDArray, NDArray]:
+    """
+    Berechnung von p (Tangente entlang konstantem u) und q (Tangente entlang konstantem v)
+    Panou, Korakitits (2020)
+    :param ell: Ellipsoid
+    :param point: Punkt
+    :return: p und q
+    """
+    n = ell.func_n(point)
+    u, v = ell.cart2para(point)
+
+    # 41-47
+    G = sqrt(1 - ell.ex ** 2 * cos(u) ** 2 - ell.ee ** 2 * sin(u) ** 2 * sin(v) ** 2)
+    q = np.array([-1 / G * sin(u) * cos(v),
+                  -1 / G * sqrt(1 - ell.ee ** 2) * sin(u) * sin(v),
+                  1 / G * sqrt(1 - ell.ex ** 2) * cos(u)])
+    p = np.array([q[1] * n[2] - q[2] * n[1],
+                  q[2] * n[0] - q[0] * n[2],
+                  q[0] * n[1] - q[1] * n[0]])
+
+    p = p / np.linalg.norm(p)
+    q = q / np.linalg.norm(q)
+
+    return p, q
+
+
+def jacobi_konstante(beta: float, omega: float, alpha: float, ell: EllipsoidTriaxial) -> float:
+    """
+    Jacobi-Konstante nach Karney (2025), Gl. (14)
+    :param beta: Beta Koordinate
+    :param omega: Omega Koordinate
+    :param alpha: Azimut alpha
+    :param ell: Ellipsoid
+    :return: Jacobi-Konstante
+    """
+    gamma_jacobi = float((ell.k ** 2) * (np.cos(beta) ** 2) * (np.sin(alpha) ** 2) - (ell.k_ ** 2) * (np.sin(omega) ** 2) * (np.cos(alpha) ** 2))
+    return gamma_jacobi
+
+
+if __name__ == "__main__":
+    ell = EllipsoidTriaxial.init_name("KarneyTest2024")
+    alpha_para = 0
+    u, v = ell.ell2para(np.pi/2, 0)
+    alpha_ell = alpha_para2ell(ell, u, v, alpha_para)
+    pass

ausgaben.py

@@ -10,7 +10,7 @@ def xyz(x: float, y: float, z: float, stellen: int) -> str:
     :param stellen: Anzahl Nachkommastellen
     :return: String zur Ausgabe der Koordinaten
     """
-    return f"""x = {(round(x,stellen))} m  y = {(round(y,stellen))} m  z = {(round(z,stellen))} m"""
+    return f"""x = {(round(x, stellen))} m  y = {(round(y, stellen))} m  z = {(round(z, stellen))} m"""
 
 
 def gms(name: str, rad: float, stellen: int) -> str:
@@ -21,5 +21,5 @@ def gms(name: str, rad: float, stellen: int) -> str:
     :param stellen: Anzahl Nachkommastellen
     :return: String zur Ausgabe des Winkels
     """
-    gms = wu.rad2gms(rad)
-    return f"{name} = {int(gms[0])}° {int(gms[1])}' {round(gms[2],stellen):.{stellen}f}''"
+    values = wu.rad2gms(rad)
+    return f"{name} = {int(values[0])}° {int(values[1])}' {round(values[2], stellen):.{stellen}f}''"

ellipsoid_triaxial.py

@@ -1,101 +1,20 @@
-import numpy as np
-from numpy import sin, cos, arctan, arctan2, sqrt, pi, arccos
-import winkelumrechnungen as wu
-import jacobian_Ligas
-import matplotlib.pyplot as plt
+import math
 from typing import Tuple
+
+import numpy as np
+from numpy import arccos, arctan, arctan2, cos, pi, sin, sqrt
 from numpy.typing import NDArray
 
+import jacobian_Ligas
+from utils_angle import wrap_mhalfpi_halfpi, wrap_mpi_pi
 
-class EllipsoidBiaxial:
-    def __init__(self, a: float, b: float):
-        self.a = a
-        self.b = b
-        self.c = a ** 2 / b
-        self.e = sqrt(a ** 2 - b ** 2) / a
-        self.e_ = sqrt(a ** 2 - b ** 2) / b
-
-    @classmethod
-    def init_name(cls, name: str):
-        if name == "Bessel":
-            a = 6377397.15508
-            b = 6356078.96290
-            return cls(a, b)
-        elif name == "Hayford":
-            a = 6378388
-            f = 1/297
-            b = a - a * f
-            return cls(a, b)
-        elif name == "Krassowski":
-            a = 6378245
-            f = 298.3
-            b = a - a * f
-            return cls(a, b)
-        elif name == "WGS84":
-            a = 6378137
-            f = 298.257223563
-            b = a - a * f
-            return cls(a, b)
-
-    @classmethod
-    def init_af(cls, a: float, f: float):
-        b = a - a * f
-        return cls(a, b)
-
-    V = lambda self, phi: sqrt(1 + self.e_ ** 2 * cos(phi) ** 2)
-    M = lambda self, phi: self.c / self.V(phi) ** 3
-    N = lambda self, phi: self.c / self.V(phi)
-
-    beta2psi = lambda self, beta: arctan(self.a / self.b * np.tan(beta))
-    beta2phi = lambda self, beta: arctan(self.a ** 2 / self.b ** 2 * np.tan(beta))
-
-    psi2beta = lambda self, psi: arctan(self.b / self.a * np.tan(psi))
-    psi2phi = lambda self, psi: arctan(self.a / self.b * np.tan(psi))
-
-    phi2beta = lambda self, phi: arctan(self.b ** 2 / self.a ** 2 * np.tan(phi))
-    phi2psi = lambda self, phi: arctan(self.b / self.a * np.tan(phi))
-
-    phi2p = lambda self, phi: self.N(phi) * cos(phi)
-
-    def cart2ell(self, Eh, Ephi, x, y, z):
-        p = sqrt(x**2+y**2)
-        # print(f"p = {round(p, 5)} m")
-
-        lamb = arctan(y/x)
-
-        phi_null = arctan(z/p*(1-self.e**2)**-1)
-
-        hi = [0]
-        phii = [phi_null]
-
-        i = 0
-
-        while True:
-            N = self.a*(1-self.e**2*sin(phii[i])**2)**(-1/2)
-            h = p/cos(phii[i])-N
-            phi = arctan(z/p*(1-(self.e**2*N)/(N+h))**(-1))
-            hi.append(h)
-            phii.append(phi)
-            dh = abs(hi[i]-h)
-            dphi = abs(phii[i]-phi)
-            i = i+1
-            if dh < Eh:
-                if dphi < Ephi:
-                    break
-        for i in range(len(phii)):
-            # print(f"P3[{i}]: {aus.gms('phi', phii[i], 5)}\th = {round(hi[i], 5)} m")
-            pass
-        return phi, lamb, h
-
-    def ell2cart(self, phi, lamb, h):
-        W = sqrt(1 - self.e**2 * sin(phi)**2)
-        N = self.a / W
-        x = (N+h) * cos(phi) * cos(lamb)
-        y = (N+h) * cos(phi) * sin(lamb)
-        z = (N * (1-self.e**2) + h) * sin(lamb)
-        return x, y, z
 
 class EllipsoidTriaxial:
+    """
+    Klasse für dreiachsige Ellipsoide
+    Parameter: Formparameter
+    Funktionen: Koordinatenumrechnungen
+    """
     def __init__(self, ax: float, ay: float, b: float):
         self.ax = ax
         self.ay = ay
@@ -109,12 +28,19 @@ class EllipsoidTriaxial:
         self.Ex = sqrt(self.ax**2 - self.b**2)
         self.Ey = sqrt(self.ay**2 - self.b**2)
         self.Ee = sqrt(self.ax**2 - self.ay**2)
+        nenner = sqrt(max(self.ax * self.ax - self.b * self.b, 0.0))
+        self.k = sqrt(max(self.ay * self.ay - self.b * self.b, 0.0)) / nenner
+        self.k_ = sqrt(max(self.ax * self.ax - self.ay * self.ay, 0.0)) / nenner
+        self.e = sqrt(max(self.ax * self.ax - self.b * self.b, 0.0)) / self.ay
 
     @classmethod
-    def init_name(cls, name: str):
+    def init_name(cls, name: str) -> EllipsoidTriaxial:
         """
-        Mögliche Ellipsoide: BursaFialova1993, BursaSima1980, Eitschberger1978, Bursa1972, Bursa1970, BesselBiaxial
+        Mögliche Ellipsoide: BursaSima1980round, KarneyTest2024, Fiction, BursaFialova1993, BursaSima1980, Eitschberger1978, Bursa1972,
+        Bursa1970
+        Panou et al (2020)
         :param name: Name des dreiachsigen Ellipsoids
+        :return: dreiachsiger Ellipsoid
         """
         if name == "BursaFialova1993":
             ax = 6378171.36
@@ -147,11 +73,6 @@ class EllipsoidTriaxial:
             ay = 6378105
             b = 6356754
             return cls(ax, ay, b)
-        elif name == "BesselBiaxial":
-            ax = 6377397.15509
-            ay = 6377397.15508
-            b = 6356078.96290
-            return cls(ax, ay, b)
         elif name == "Fiction":
             ax = 6000000
             ay = 4000000
@@ -162,6 +83,8 @@ class EllipsoidTriaxial:
             ay = 1
             b = 1 / sqrt(2)
             return cls(ax, ay, b)
+        else:
+            raise Exception(f"EllipsoidTriaxial.init_name: Name {name} unbekannt")
 
     def func_H(self, point: NDArray) -> float:
         """
@@ -201,7 +124,12 @@ class EllipsoidTriaxial:
         c0 = (self.ax ** 2 * self.ay ** 2 + self.ax ** 2 * self.b ** 2 + self.ay ** 2 * self.b ** 2 -
               (self.ay ** 2 + self.b ** 2) * x ** 2 - (self.ax ** 2 + self.b ** 2) * y ** 2 - (
                       self.ax ** 2 + self.ay ** 2) * z ** 2)
-        t2 = (-c1 + sqrt(c1 ** 2 - 4 * c0)) / 2
+        if c1 ** 2 - 4 * c0 < -1e-9:
+            raise Exception("t1, t2: Negativer Wurzelterm")
+        elif c1 ** 2 - 4 * c0 < 0:
+            t2 = 0
+        else:
+            t2 = (-c1 + sqrt(c1 ** 2 - 4 * c0)) / 2
         if t2 == 0:
             t2 = 1e-18
         t1 = c0 / t2
@@ -210,12 +138,9 @@ class EllipsoidTriaxial:
     def ellu2cart(self, beta: float, lamb: float, u: float) -> NDArray:
         """
         Panou 2014 12ff.
-        Elliptische Breite+Länge sind nicht gleich der geodätischen
-        Verhältnisse des Ellipsoids bekannt, Größe verändern bis Punkt erreicht,
-           dann ist u die Größe entlang der z-Achse
-        :param beta: ellipsoidische Breite [rad]
-        :param lamb: ellipsoidische Länge [rad]
-        :param u: Größe entlang der z-Achse
+        :param beta: ellipsoidische Breite
+        :param lamb: ellipsoidische Länge
+        :param u: radiale Koordinate entlang der kleinen Halbachse
         :return: Punkt in kartesischen Koordinaten
         """
         x = sqrt(u**2 + self.Ex**2) * sqrt(cos(beta)**2 + self.Ee**2/self.Ex**2 * sin(beta)**2) * cos(lamb)
@@ -228,7 +153,7 @@ class EllipsoidTriaxial:
         """
         Panou 2014 15ff.
         :param point: Punkt in kartesischen Koordinaten
-        :return: elliptische Breite, elliptische Länge, Größe entlang der z-Achse
+        :return: ellipsoidische Breite, ellipsoidische Länge, radiale Koordinate entlang der kleinen Halbachse
         """
         x, y, z = point
         c2 = self.ax**2 + self.ay**2 + self.b**2 - x**2 - y**2 - z**2
@@ -244,7 +169,6 @@ class EllipsoidTriaxial:
         s1 = 2 * sqrt(p) * cos(omega/3) - c2/3
         s2 = 2 * sqrt(p) * cos(omega/3 - 2*pi/3) - c2/3
         s3 = 2 * sqrt(p) * cos(omega/3 - 4*pi/3) - c2/3
-        # print(s1, s2, s3)
 
         beta = arctan(sqrt((-self.b**2 - s2) / (self.ay**2 + s2)))
         if abs((-self.ay**2 - s3) / (self.ax**2 + s3)) > 1e-7:
@@ -258,8 +182,8 @@ class EllipsoidTriaxial:
     def ell2cart(self, beta: float | NDArray, lamb: float | NDArray) -> NDArray:
         """
         Panou, Korakitis 2019 2
-        :param beta: elliptische Breite [rad]
-        :param lamb: elliptische Länge [rad]
+        :param beta: ellipsoidische Breite
+        :param lamb: ellipsoidische Länge
         :return: Punkt in kartesischen Koordinaten
         """
         beta = np.asarray(beta, dtype=float)
@@ -267,6 +191,11 @@ class EllipsoidTriaxial:
 
         beta, lamb = np.broadcast_arrays(beta, lamb)
 
+        beta = np.where(
+            np.isclose(np.abs(beta), pi / 2, atol=1e-15),
+            beta * 8999999999999999 / 9000000000000000,
+            beta
+        )
         B = self.Ex ** 2 * cos(beta) ** 2 + self.Ee ** 2 * sin(beta) ** 2
         L = self.Ex ** 2 - self.Ee ** 2 * cos(lamb) ** 2
 
@@ -294,8 +223,8 @@ class EllipsoidTriaxial:
     def ell2cart_bektas(self, beta: float | NDArray, omega: float | NDArray) -> NDArray:
         """
         Bektas 2015
-        :param beta: elliptische Breite [rad]
-        :param omega: elliptische Länge [rad]
+        :param beta: ellipsoidische Breite
+        :param omega: ellipsoidische Länge
         :return: Punkt in kartesischen Koordinaten
         """
         x = self.ax * cos(omega) * sqrt((self.ax**2 - self.ay**2 * sin(beta)**2 - self.b**2 * cos(beta)**2) / (self.ax**2 - self.b**2))
@@ -307,8 +236,8 @@ class EllipsoidTriaxial:
     def ell2cart_karney(self, beta: float | NDArray, lamb: float | NDArray) -> NDArray:
         """
         Karney 2025 Geographic Lib
-        :param beta: elliptische Breite [rad]
-        :param lamb: elliptische Länge [rad]
+        :param beta: ellipsoidische Breite
+        :param lamb: ellipsoidische Länge
         :return: Punkt in kartesischen Koordinaten
         """
         k = sqrt(self.ay**2 - self.b**2) / sqrt(self.ax**2 - self.b**2)
@@ -318,60 +247,105 @@ class EllipsoidTriaxial:
         Z = self.b * sin(beta) * sqrt(k**2 + k_**2 * sin(lamb)**2)
         return np.array([X, Y, Z])
 
-    def cart2ell(self, point: NDArray, eps: float = 1e-12, maxI: int = 100) -> Tuple[float, float]:
+    def cart2ell_yFake(self, point: NDArray, delta_y: float = 1e-4) -> Tuple[float, float]:
+        """
+        Bei Fehlschlagen von cart2ell
+        :param point: Punkt in kartesischen Koordinaten
+        :param delta_y: Startwert für Suche nach kleinstmöglichem delta_y
+        :return: ellipsoidische Breite und Länge
+        """
+        x, y, z = point
+        best_delta = np.inf
+        while True:
+            try:
+                y1 = y - delta_y
+                beta1, lamb1 = self.cart2ell(np.array([x, y1, z]), noFake=True)
+                point1 = self.ell2cart(beta1, lamb1)
+
+                y2 = y + delta_y
+                beta2, lamb2 = self.cart2ell(np.array([x, y2, z]), noFake=True)
+                point2 = self.ell2cart(beta2, lamb2)
+
+                pointM = (point1 + point2) / 2
+
+                actual_delta = np.linalg.norm(point - pointM)
+            except:
+                actual_delta = np.inf
+
+            if actual_delta < best_delta:
+                best_delta = actual_delta
+                delta_y /= 10
+            else:
+                delta_y *= 10
+
+                y1 = y - delta_y
+                beta1, lamb1 = self.cart2ell(np.array([x, y1, z]), noFake=True)
+
+                return beta1, lamb1
+
+    def cart2ell(self, point: NDArray, eps: float = 1e-12, maxI: int = 100, noFake: bool = False) -> Tuple[float, float]:
         """
         Panou, Korakitis 2019 3f. (num)
         :param point:  Punkt in kartesischen Koordinaten
         :param eps: zu erreichende Genauigkeit
         :param maxI: maximale Anzahl Iterationen
-        :return: elliptische Breite und Länge [rad]
+        :param noFake: y numerisch anpassen?
+        :return: ellipsoidische Breite und Länge
         """
         x, y, z = point
         beta, lamb = self.cart2ell_panou(point)
         delta_ell = np.array([np.inf, np.inf]).T
+        tiny = 1e-30
 
-        i = 0
-        while np.sum(delta_ell) > eps and i < maxI:
-            x0, y0, z0 = self.ell2cart(beta, lamb)
-            delta_l = np.array([x-x0, y-y0, z-z0]).T
+        try:
+            i = 0
+            while np.linalg.norm(delta_ell) > eps and i < maxI:
+                x0, y0, z0 = self.ell2cart(beta, lamb)
+                delta_l = np.array([x-x0, y-y0, z-z0]).T
 
-            B = self.Ex ** 2 * cos(beta) ** 2 + self.Ee ** 2 * sin(beta) ** 2
-            L = self.Ex ** 2 - self.Ee ** 2 * cos(lamb) ** 2
+                B = max(self.Ex ** 2 * cos(beta) ** 2 + self.Ee ** 2 * sin(beta) ** 2, tiny)
+                L = max(self.Ex ** 2 - self.Ee ** 2 * cos(lamb) ** 2, tiny)
 
-            J = np.array([[(-self.ax * self.Ey ** 2) / (2 * self.Ex) * sin(2 * beta) / sqrt(B) * cos(lamb),
-                           -self.ax / self.Ex * sqrt(B) * sin(lamb)],
-                          [-self.ay * sin(beta) * sin(lamb),
-                           self.ay * cos(beta) * cos(lamb)],
-                          [self.b / self.Ex * cos(beta) * sqrt(L),
-                           (self.b * self.Ee ** 2) / (2 * self.Ex) * sin(beta) * sin(2 * lamb) / sqrt(L)]])
+                J = np.array([[(-self.ax * self.Ey ** 2) / (2 * self.Ex) * sin(2 * beta) / sqrt(B) * cos(lamb),
+                               -self.ax / self.Ex * sqrt(B) * sin(lamb)],
+                              [-self.ay * sin(beta) * sin(lamb),
+                               self.ay * cos(beta) * cos(lamb)],
+                              [self.b / self.Ex * cos(beta) * sqrt(L),
+                               (self.b * self.Ee ** 2) / (2 * self.Ex) * sin(beta) * sin(2 * lamb) / sqrt(L)]])
 
-            N = J.T @ J
-            det = N[0, 0] * N[1, 1] - N[0, 1] * N[1, 0]
-            if abs(det) < eps:
-                det = eps
-            N_inv = 1 / det * np.array([[N[1, 1], -N[0, 1]], [-N[1, 0], N[0, 0]]])
-            delta_ell = N_inv @ J.T @ delta_l
-            beta += delta_ell[0]
-            lamb += delta_ell[1]
-            i += 1
+                N = J.T @ J
+                det = N[0, 0] * N[1, 1] - N[0, 1] * N[1, 0]
+                N_inv = 1 / det * np.array([[N[1, 1], -N[0, 1]], [-N[1, 0], N[0, 0]]])
+                delta_ell = N_inv @ J.T @ delta_l
+                # delta_ell, *_ = np.linalg.lstsq(J, delta_l, rcond=None)
 
-        if i == maxI:
-            raise Exception("Umrechung ist nicht konvergiert")
+                beta += delta_ell[0]
+                lamb += delta_ell[1]
+                i += 1
 
-        point_n = self.ell2cart(beta, lamb)
-        delta_r = np.linalg.norm(point - point_n, axis=-1)
+            if i == maxI:
+                raise Exception("Umrechnung cart2ell: nicht konvergiert")
 
-        if delta_r > 1e-4:
-            # raise Exception("Fehler in der Umrechnung cart2ell")
-            print(f"Fehler in der Umrechnung cart2ell, deltaR = {delta_r}m")
+            point_n = self.ell2cart(beta, lamb)
+            delta_r = np.linalg.norm(point - point_n, axis=-1)
+            if delta_r > 1e-6:
+                raise Exception("Umrechnung cart2ell: Punktdifferenz")
 
-        return beta, lamb
+            return wrap_mhalfpi_halfpi(beta), wrap_mpi_pi(lamb)
+
+        except Exception as e:
+            # Wenn die Berechnung fehlschlägt auf Grund von sehr kleinem y, solange anpassen, bis Umrechnung ohne Fehler
+            delta_y = 10 ** math.floor(math.log10(abs(self.ay/1000)))
+            if abs(y) < delta_y and not noFake:
+                return self.cart2ell_yFake(point, delta_y)
+            else:
+                raise e
 
     def cart2ell_panou(self, point: NDArray) -> Tuple[float, float]:
         """
         Panou, Korakitis 2019 2f. (analytisch -> Näherung)
         :param point: Punkt in kartesischen Koordinaten
-        :return: elliptische Breite, elliptische Länge
+        :return: ellipsoidische Breite und Länge
         """
         x, y, z = point
 
@@ -397,9 +371,9 @@ class EllipsoidTriaxial:
         t1, t2 = self.func_t12(point)
 
         num_beta = max(t1 - self.b ** 2, 0)
-        den_beta = max(self.ay ** 2 - t1, 0)
+        den_beta = max(self.ay ** 2 - t1, 1e-30)
         num_lamb = max(t2 - self.ay ** 2, 0)
-        den_lamb = max(self.ax ** 2 - t2, 0)
+        den_lamb = max(self.ax ** 2 - t2, 1e-30)
 
         beta = arctan(sqrt(num_beta / den_beta))
         lamb = arctan(sqrt(num_lamb / den_lamb))
@@ -430,7 +404,7 @@ class EllipsoidTriaxial:
         :param point:  Punkt in kartesischen Koordinaten
         :param eps: zu erreichende Genauigkeit
         :param maxI: maximale Anzahl Iterationen
-        :return: elliptische Breite und Länge [rad]
+        :return: ellipsoidische Breite und Länge
         """
         x, y, z = point
         phi, lamb = self.cart2para(point)
@@ -449,16 +423,16 @@ class EllipsoidTriaxial:
             i += 1
 
         if i == maxI:
-            raise Exception("Umrechung ist nicht konvergiert")
+            raise Exception("Umrechnung cart2ell: nicht konvergiert")
 
         return phi, lamb
 
     def geod2cart(self, phi: float | NDArray, lamb: float | NDArray, h: float) -> NDArray:
         """
         Ligas 2012, 250
-        :param phi: geodätische Breite [rad]
-        :param lamb: geodätische Länge [rad]
-        :param h: Höhe über dem Ellipsoid
+        :param phi: geodätische Breite
+        :param lamb: geodätische Länge
+        :param h: geodätische Höhe
         :return: kartesische Koordinaten
         """
         v = self.ax / sqrt(1 - self.ex**2*sin(phi)**2-self.ee**2*cos(phi)**2*sin(lamb)**2)
@@ -474,7 +448,7 @@ class EllipsoidTriaxial:
         :param point: Punkt in kartesischen Koordinaten
         :param maxIter: maximale Anzahl Iterationen
         :param maxLoa: Level of Accuracy, das erreicht werden soll
-        :return: phi, lambda, h
+        :return: geodätische Breite, Länge, Höhe
         """
         xG, yG, zG = point
 
@@ -515,32 +489,44 @@ class EllipsoidTriaxial:
                 invJ, fxE = jacobian_Ligas.case2(E, F, G, np.array([xG, yG, zG]), pE)
             elif mode == "ligas3":
                 invJ, fxE = jacobian_Ligas.case3(E, F, G, np.array([xG, yG, zG]), pE)
+            else:
+                raise Exception(f"cart2geod: Modus {mode} nicht bekannt")
             pEi = pE.reshape(-1, 1) - invJ @ fxE.reshape(-1, 1)
             pEi = pEi.reshape(1, -1).flatten()
             loa = sqrt((pEi[0]-pE[0])**2 + (pEi[1]-pE[1])**2 + (pEi[2]-pE[2])**2)
             pE = pEi
             i += 1
 
-        phi = arctan((1-self.ee**2) / (1-self.ex**2) * pE[2] / sqrt((1-self.ee**2)**2 * pE[0]**2 + pE[1]**2))
-        lamb = arctan(1/(1-self.ee**2) * pE[1]/pE[0])
-        h = np.sign(zG - pE[2]) * np.sign(pE[2]) * sqrt((pE[0] - xG) ** 2 + (pE[1] - yG) ** 2 + (pE[2] - zG) ** 2)
+        if i == maxIter and loa > maxLoa:
+            act_mode = int(mode[-1])
+            new_mode = 3 if act_mode == 1 else act_mode - 1
+            phi, lamb, h = self.cart2geod(point, f"ligas{new_mode}", maxIter, maxLoa)
 
-        if xG < 0 and yG < 0:
-            lamb = -pi + lamb
+        else:
+            phi = arctan((1-self.ee**2) / (1-self.ex**2) * pE[2] / sqrt((1-self.ee**2)**2 * pE[0]**2 + pE[1]**2))
+            lamb = arctan(1/(1-self.ee**2) * pE[1]/pE[0])
+            h = np.sign(zG - pE[2]) * np.sign(pE[2]) * sqrt((pE[0] - xG) ** 2 + (pE[1] - yG) ** 2 + (pE[2] - zG) ** 2)
+            if h < -self.ax:
+                act_mode = int(mode[-1])
+                new_mode = 3 if act_mode == 1 else act_mode - 1
+                phi, lamb, h = self.cart2geod(point, f"ligas{new_mode}", maxIter, maxLoa)
+            else:
+                if xG < 0 and yG < 0:
+                    lamb += -pi
 
-        elif xG < 0:
-            lamb = pi + lamb
+                elif xG < 0:
+                    lamb += pi
 
-        if abs(zG) < eps:
-            phi = 0
-
-        return phi, lamb, h
+                if abs(zG) < eps:
+                    phi = 0
+        wrap_mhalfpi_halfpi(phi), wrap_mpi_pi(lamb)
+        return wrap_mhalfpi_halfpi(phi), wrap_mpi_pi(lamb), h
 
     def para2cart(self, u: float | NDArray, v: float | NDArray) -> NDArray:
         """
         Panou, Korakitits 2020, 4
-        :param u: Parameter u
-        :param v: Parameter v
+        :param u: parametrische Breite
+        :param v: parametrische Länge
         :return: Punkt in kartesischen Koordinaten
         """
         x = self.ax * cos(u) * cos(v)
@@ -553,7 +539,7 @@ class EllipsoidTriaxial:
         """
         Panou, Korakitits 2020, 4
         :param point: Punkt in kartesischen Koordinaten
-        :return: parametrische Koordinaten
+        :return: parametrische Breite, Länge
         """
         x, y, z = point
 
@@ -571,25 +557,25 @@ class EllipsoidTriaxial:
             v = 2 * arctan2(v_check1, v_check2 + v_factor)
         else:
             v = pi/2 - 2 * arctan2(v_check2, v_check1 + v_factor)
-
-        return u, v
+        wrap_mhalfpi_halfpi(u), wrap_mpi_pi(v)
+        return wrap_mhalfpi_halfpi(u), wrap_mpi_pi(v)
 
     def ell2para(self, beta: float, lamb: float) -> Tuple[float, float]:
         """
-        Umrechung von elliptischen in parametrische Koordinaten (über kartesische Koordinaten)
-        :param beta: elliptische Breite
-        :param lamb: elliptische Länge
-        :return: parametrische Koordinaten
+        Umrechung von ellipsoidischen in parametrische Koordinaten (über kartesische Koordinaten)
+        :param beta: ellipsoidische Breite
+        :param lamb: ellipsoidische Länge
+        :return: parametrische Breite, Länge
         """
         cart = self.ell2cart(beta, lamb)
         return self.cart2para(cart)
 
     def para2ell(self, u: float, v: float) -> Tuple[float, float]:
         """
-        Umrechung von parametrischen in elliptische Koordinaten (über kartesische Koordinaten)
-        :param u: u
-        :param v: v
-        :return: elliptische Koordinaten
+        Umrechung von parametrischen in ellipsoidische Koordinaten (über kartesische Koordinaten)
+        :param u: parametrische Breite
+        :param v: parametrische Länge
+        :return: ellipsoidische Breite, Länge
         """
         cart = self.para2cart(u, v)
         return self.cart2ell(cart)
@@ -597,12 +583,12 @@ class EllipsoidTriaxial:
     def para2geod(self, u: float, v: float, mode: str = "ligas3", maxIter: int = 30, maxLoa: float = 0.005) -> Tuple[float, float, float]:
         """
         Umrechung von parametrischen in geodätische Koordinaten (über kartesische Koordinaten)
-        :param u: u
-        :param v: v
+        :param u: parametrische Breite
+        :param v: parametrische Länge
         :param mode: ligas1, ligas2, oder ligas3
         :param maxIter: maximale Anzahl Iterationen
         :param maxLoa: Level of Accuracy, das erreicht werden soll
-        :return: geodätische Koordinaten
+        :return: geodätische Breite, Länge, Höhe
         """
         cart = self.para2cart(u, v)
         return self.cart2geod(cart, mode, maxIter, maxLoa)
@@ -610,41 +596,41 @@ class EllipsoidTriaxial:
     def geod2para(self, phi: float, lamb: float, h: float) -> Tuple[float, float]:
         """
         Umrechung von geodätischen in parametrische Koordinaten (über kartesische Koordinaten)
-        :param phi: u
-        :param lamb: v
+        :param phi: geodätische Breite
+        :param lamb: geodätische Länge
         :param h: geodätische Höhe
-        :return: parametrische Koordinaten
+        :return: parametrische Breite, Länge
         """
         cart = self.geod2cart(phi, lamb, h)
         return self.cart2para(cart)
 
     def ell2geod(self, beta: float, lamb: float, mode: str = "ligas3", maxIter: int = 30, maxLoa: float = 0.005) -> Tuple[float, float, float]:
         """
-        Umrechung von elliptischen in geodätische Koordinaten (über kartesische Koordinaten)
-        :param beta: elliptische Breite
-        :param lamb: eliptische Länge
+        Umrechung von ellipsoidischen in geodätische Koordinaten (über kartesische Koordinaten)
+        :param beta: ellipsoidische Breite
+        :param lamb: ellipsoidische Länge
         :param mode: ligas1, ligas2, oder ligas3
         :param maxIter: maximale Anzahl Iterationen
         :param maxLoa: Level of Accuracy, das erreicht werden soll
-        :return: geodätische Koordinaten
+        :return: geodätische Breite, Länge, Höhe
         """
         cart = self.ell2cart(beta, lamb)
         return self.cart2geod(cart, mode, maxIter, maxLoa)
 
     def geod2ell(self, phi: float, lamb: float, h: float) -> Tuple[float, float]:
         """
-        Umrechung von geodätischen in elliptische Koordinaten (über kartesische Koordinaten)
-        :param phi: u
-        :param lamb: v
+        Umrechung von geodätischen in ellipsoidische Koordinaten (über kartesische Koordinaten)
+        :param phi: geodätische Breite
+        :param lamb: geodätische Länge
         :param h: geodätische Höhe
-        :return: elliptische Koordinaten
+        :return: ellipsoidische Breite, Länge
         """
         cart = self.geod2cart(phi, lamb, h)
         return self.cart2ell(cart)
 
     def point_on(self, point: NDArray) -> bool:
         """
-        Test, ob ein Punkt auf dem Ellipsoid liegt.
+        Test, ob ein Punkt auf dem Ellipsoid liegt
         :param point: kartesische 3D-Koordinaten
         :return: Punkt auf dem Ellispoid?
         """
@@ -677,62 +663,71 @@ class EllipsoidTriaxial:
 
 
 if __name__ == "__main__":
-    ell = EllipsoidTriaxial.init_name("BursaSima1980round")
-    diff_list = []
-    diffs_para = []
-    diffs_ell = []
-    diffs_geod = []
-    points = []
-    for v_deg in range(-180, 181, 5):
-        for u_deg in range(-90, 91, 5):
-            v = wu.deg2rad(v_deg)
-            u = wu.deg2rad(u_deg)
-            point = ell.para2cart(u, v)
-            points.append(point)
+    ell = EllipsoidTriaxial.init_name("KarneyTest2024")
+    # cart = ell.ell2cart(pi/2, 0)
+    # print(cart)
+    # cart = ell.ell2cart(pi/2*8999999999999999/9000000000000000, 0)
+    # print(cart)
+    elli = ell.cart2ell(np.array([0, 0.0, 1/sqrt(2)]))
+    print(elli)
 
-            elli = ell.cart2ell(point)
-            cart_elli = ell.ell2cart(elli[0], elli[1])
-            diff_ell = np.linalg.norm(point - cart_elli, axis=-1)
-
-            para = ell.cart2para(point)
-            cart_para = ell.para2cart(para[0], para[1])
-            diff_para = np.linalg.norm(point - cart_para, axis=-1)
-
-            geod = ell.cart2geod(point, "ligas3")
-            cart_geod = ell.geod2cart(geod[0], geod[1], geod[2])
-            diff_geod3 = np.linalg.norm(point - cart_geod, axis=-1)
-
-            diff_list.append([v_deg, u_deg, diff_ell, diff_para, diff_geod3])
-            diffs_ell.append([diff_ell])
-            diffs_para.append([diff_para])
-            diffs_geod.append([diff_geod3])
-
-    diff_list = np.array(diff_list)
-    diffs_ell = np.array(diffs_ell)
-    diffs_para = np.array(diffs_para)
-    diffs_geod = np.array(diffs_geod)
-
-    pass
-    points = np.array(points)
-    fig = plt.figure()
-    ax = fig.add_subplot(projection='3d')
-
-    sc = ax.scatter(
-        points[:, 0],
-        points[:, 1],
-        points[:, 2],
-        c=diffs_ell,  # Farbcode = diff
-        cmap='viridis',  # Colormap
-        s=10 + 20 * diffs_ell,  # optional: Größe abhängig vom diff
-        alpha=0.8
-    )
-
-    # Farbskala
-    cbar = plt.colorbar(sc)
-    cbar.set_label("diff")
-
-    ax.set_xlabel("X")
-    ax.set_ylabel("Y")
-    ax.set_zlabel("Z")
-
-    plt.show()
+    # ell = EllipsoidTriaxial.init_name("BursaSima1980")
+    # diff_list = []
+    # diffs_para = []
+    # diffs_ell = []
+    # diffs_geod = []
+    # points = []
+    # for v_deg in range(-180, 181, 5):
+    #     for u_deg in range(-90, 91, 5):
+    #         v = wu.deg2rad(v_deg)
+    #         u = wu.deg2rad(u_deg)
+    #         point = ell.para2cart(u, v)
+    #         points.append(point)
+    #
+    #         elli = ell.cart2ell(point)
+    #         cart_elli = ell.ell2cart(elli[0], elli[1])
+    #         diff_ell = np.linalg.norm(point - cart_elli, axis=-1)
+    #
+    #         para = ell.cart2para(point)
+    #         cart_para = ell.para2cart(para[0], para[1])
+    #         diff_para = np.linalg.norm(point - cart_para, axis=-1)
+    #
+    #         geod = ell.cart2geod(point, "ligas3")
+    #         cart_geod = ell.geod2cart(geod[0], geod[1], geod[2])
+    #         diff_geod3 = np.linalg.norm(point - cart_geod, axis=-1)
+    #
+    #         diff_list.append([v_deg, u_deg, diff_ell, diff_para, diff_geod3])
+    #         diffs_ell.append([diff_ell])
+    #         diffs_para.append([diff_para])
+    #         diffs_geod.append([diff_geod3])
+    #
+    # diff_list = np.array(diff_list)
+    # diffs_ell = np.array(diffs_ell)
+    # diffs_para = np.array(diffs_para)
+    # diffs_geod = np.array(diffs_geod)
+    #
+    # pass
+    #
+    # points = np.array(points)
+    # fig = plt.figure()
+    # ax = fig.add_subplot(projection='3d')
+    #
+    # sc = ax.scatter(
+    #     points[:, 0],
+    #     points[:, 1],
+    #     points[:, 2],
+    #     c=diffs_ell,  # Farbcode = diff
+    #     cmap='viridis',  # Colormap
+    #     s=10 + 20 * diffs_ell,  # optional: Größe abhängig vom diff
+    #     alpha=0.8
+    # )
+    #
+    # # Farbskala
+    # cbar = plt.colorbar(sc)
+    # cbar.set_label("diff")
+    #
+    # ax.set_xlabel("X")
+    # ax.set_ylabel("Y")
+    # ax.set_zlabel("Z")
+    #
+    # plt.show()

jacobian_Ligas.py

@@ -1,6 +1,19 @@
-import numpy as np
+from typing import Tuple
 
-def case1(E: float, F: float, G: float, pG: np.ndarray, pE: np.ndarray):
+import numpy as np
+from numpy.typing import NDArray
+
+
+def case1(E: float, F: float, G: float, pG: NDArray, pE: NDArray) -> Tuple[NDArray, NDArray]:
+    """
+    Aufstellen des Gleichungssystem für den ersten Fall
+    :param E: Konstante E
+    :param F: Konstante F
+    :param G: Konstante G
+    :param pG: Punkt über dem Ellipsoid
+    :param pE: Punkt auf dem Ellipsoid
+    :return: inverse Jacobi-Matrix, Gleichungssystem
+    """
     j11 = 2 * E * pE[0]
     j12 = 2 * F * pE[1]
     j13 = 2 * G * pE[2]
@@ -23,7 +36,16 @@ def case1(E: float, F: float, G: float, pG: np.ndarray, pE: np.ndarray):
 
     return invJ, fxE
 
-def case2(E: float, F: float, G: float, pG: np.ndarray, pE: np.ndarray):
+def case2(E: float, F: float, G: float, pG: NDArray, pE: NDArray) -> Tuple[NDArray, NDArray]:
+    """
+    Aufstellen des Gleichungssystem für den zweiten Fall
+    :param E: Konstante E
+    :param F: Konstante F
+    :param G: Konstante G
+    :param pG: Punkt über dem Ellipsoid
+    :param pE: Punkt auf dem Ellipsoid
+    :return: inverse Jacobi-Matrix, Gleichungssystem
+    """
     j11 = 2 * E * pE[0]
     j12 = 2 * F * pE[1]
     j13 = 2 * G * pE[2]
@@ -48,7 +70,16 @@ def case2(E: float, F: float, G: float, pG: np.ndarray, pE: np.ndarray):
 
     return invJ, fxE
 
-def case3(E: float, F: float, G: float, pG: np.ndarray, pE: np.ndarray):
+def case3(E: float, F: float, G: float, pG: NDArray, pE: NDArray) -> Tuple[NDArray, NDArray]:
+    """
+    Aufstellen des Gleichungssystem für den dritten Fall
+    :param E: Konstante E
+    :param F: Konstante F
+    :param G: Konstante G
+    :param pG: Punkt über dem Ellipsoid
+    :param pE: Punkt auf dem Ellipsoid
+    :return: inverse Jacobi-Matrix, Gleichungssystem
+    """
     j11 = 2 * E * pE[0]
     j12 = 2 * F * pE[1]
     j13 = 2 * G * pE[2]

kugel.py

@@ -1,78 +0,0 @@
-import numpy as np
-
-
-def cart2sph(x, y, z):
-    r = np.sqrt(x**2 + y**2 + z**2)
-    phi = np.atan2(z, np.sqrt(x**2 + y**2))
-    lamb = np.atan2(y, x)
-
-    return r, np.rad2deg(phi), np.rad2deg(lamb)
-
-
-def sph2cart(r, phi, lamb):
-    phi_rad = np.deg2rad(phi)
-    lamb_rad = np.deg2rad(lamb)
-
-    x = r * np.cos(phi_rad) * np.cos(lamb_rad)
-    y = r * np.cos(phi_rad) * np.sin(lamb_rad)
-    z = r * np.sin(phi_rad)
-
-    return x, y, z
-
-
-def kugel_erste_gha(R, phi_1, lamb_1, s_d, a_12):
-
-    s = s_d / R
-    lamb_1_rad = np.deg2rad(lamb_1)
-    phi_1_rad = np.deg2rad(phi_1)
-    a_12_rad = np.deg2rad(a_12)
-
-    sin_satz = np.sin(s) * np.sin(a_12_rad)
-    sin_kos = np.cos(phi_1_rad) * np.cos(s) - np.sin(phi_1_rad) * np.sin(s) * np.cos(a_12_rad)
-    delta_lam = np.atan2(sin_satz, sin_kos)
-    lamb_2_rad = lamb_1_rad + delta_lam
-
-    phi_2_rad = np.asin(np.sin(phi_1_rad) * np.cos(s) + np.cos(phi_1_rad) * np.sin(s) * np.cos(a_12_rad))
-
-    sin_satz_2 = -np.cos(phi_1_rad) * np.sin(a_12_rad)
-    sin_kos_2 = np.sin(s) * np.sin(phi_1_rad) - np.cos(s) * np.cos(phi_1_rad) * np.cos(a_12_rad)
-    a_21_rad = np.atan2(sin_satz_2, sin_kos_2)
-    if a_21_rad < 0:
-        a_21_rad += 2 * np.pi
-
-    return np.rad2deg(phi_2_rad), np.rad2deg(lamb_2_rad), np.rad2deg(a_21_rad)
-
-
-def kugel_zweite_gha(R, phi_1, lamb_1, phi_2, lamb_2):
-    phi_1_rad = np.deg2rad(phi_1)
-    lamb_1_rad = np.deg2rad(lamb_1)
-    phi_2_rad = np.deg2rad(phi_2)
-    lamb_2_rad = np.deg2rad(lamb_2)
-
-    sin_satz_1 = np.cos(phi_2_rad) * np.sin(lamb_2_rad - lamb_1_rad)
-    sin_kos_1 = np.cos(phi_1_rad) * np.sin(phi_2_rad) - np.sin(phi_1_rad) * np.cos(phi_2_rad) * np.cos(lamb_2_rad - lamb_1_rad)
-    a_12_rad = np.atan2(sin_satz_1, sin_kos_1)
-
-    kos = np.sin(phi_1_rad) * np.sin(phi_2_rad) + np.cos(phi_1_rad) * np.cos(phi_2_rad) * np.cos(lamb_2_rad - lamb_1_rad)
-    s = np.acos(kos)
-    s_d = R * s
-
-    sin_satz_2 = -np.cos(phi_1_rad) * np.sin(lamb_2_rad - lamb_1_rad)
-    sin_kos_2 = np.cos(phi_2_rad) * np.sin(phi_1_rad) - np.sin(phi_2_rad) * np.cos(phi_1_rad) * np.cos(lamb_2_rad - lamb_1_rad)
-    a_21_rad = np.atan2(sin_satz_2, sin_kos_2)
-    if a_21_rad < 0:
-        a_21_rad += 2 * np.pi
-
-    return s_d, np.rad2deg(a_12_rad), np.rad2deg(a_21_rad)
-
-
-if __name__ == "__main__":
-    R = 6378815.904                                             # Bern
-    phi_1 = 10
-    lamb_1 = 40
-    a_12 = 100
-    s = 10000
-    phi_2, lamb_2, _ = kugel_erste_gha(R, phi_1, lamb_1, s, a_12)
-
-    print(kugel_erste_gha(R, phi_1, lamb_1, s, a_12))
-    print(kugel_zweite_gha(R, phi_1, lamb_1, phi_2, lamb_2))

nicht abgeben/GHA_biaxial/bessel.py

@@ -0,0 +1,36 @@
+from typing import Tuple
+
+import scipy as sp
+from ellipsoid_biaxial import EllipsoidBiaxial
+from numpy import *
+
+
+def gha1(re: EllipsoidBiaxial, phi0: float, lamb0: float, alpha0:float, s: float) -> Tuple[float, float, float]:
+    """
+    Berechnung der 1.GHA auf einem Rotationsellipsoid nach Bessel
+    :param re:
+    :param phi0:
+    :param lamb0:
+    :param alpha0:
+    :param s:
+    :return:
+    """
+    psi0 = re.phi2psi(phi0)
+    clairant = arcsin(cos(psi0) * sin(alpha0))
+    sigma0 = arcsin(sin(psi0) / cos(clairant))
+
+    sqrt_sigma = lambda sigma: sqrt(1 + re.e_ ** 2 * cos(clairant) ** 2 * sin(sigma) ** 2)
+    int_sqrt_sigma = lambda sigma: sp.integrate.quad(sqrt_sigma, sigma0, sigma)[0]
+
+    f_sigma1_i = lambda sigma1_i: (int_sqrt_sigma(sigma1_i) - s / re.b)
+
+    sigma1_0 = sigma0 + s / re.a
+    sigma1 = sp.optimize.newton(f_sigma1_i, sigma1_0)
+    psi1 = arcsin(cos(clairant) * sin(sigma1))
+    phi1 = re.psi2phi(psi1)
+    alpha1 = arcsin(sin(clairant) / cos(psi1))
+
+    f_d_lambda = lambda sigma: sin(clairant) * sqrt_sigma(sigma) / (1 - cos(clairant) ** 2 * sin(sigma) ** 2)
+    d_lambda = sqrt(1-re.e**2) * sp.integrate.quad(f_d_lambda, sigma0, sigma1)[0]
+    lamb1 = lamb0 + d_lambda
+    return phi1, lamb1, alpha1

nicht abgeben/GHA_biaxial/gauss.py

@@ -0,0 +1,101 @@
+from typing import Tuple
+
+from ellipsoid_biaxial import EllipsoidBiaxial
+from numpy import arctan, cos, sin, sqrt, tan
+
+
+def gha1(re: EllipsoidBiaxial, phi0: float, lamb0: float, alpha0: float, s: float, eps: float = 1e-12) -> Tuple[float, float, float]:
+    """
+    Berechnung der 1. Geodätische Hauptaufgabe nach Gauß´schen Mittelbreitenformeln
+    :param re: Klasse Ellipsoid
+    :param phi0: Breite Punkt 0
+    :param lamb0: Länge Punkt 0
+    :param alpha0: Azimut der geodätischen Linie in Punkt 1
+    :param s: Strecke zu Punkt 1
+    :param eps: Abbruchkriterium für Winkelgrößen
+    :return: Breite, Länge, Azimut von Punkt 21
+    """
+    t = lambda phi: tan(phi)
+    eta = lambda phi: sqrt(re.e_ ** 2 * cos(phi) ** 2)
+
+    F1 = lambda alpha, phi, s: 1 + (2 + 3 * t(phi) ** 2 + 2 * eta(phi) ** 2) / (24 * re.N(phi) ** 2) * sin(alpha) ** 2 * s ** 2 \
+                               + (t(phi)**2 - 1) * eta(phi) ** 2 / (8 * re.N(phi) ** 2) * cos(alpha) ** 2 * s ** 2
+
+    F2 = lambda alpha, phi, s: 1 + t(phi) ** 2 / (24 * re.N(phi) ** 2) * sin(alpha) ** 2 * s ** 2 \
+                               - (1 + eta(phi)**2 - 9 * eta(phi)**2 * t(phi)**2) / (24 * re.N(phi) ** 2) * cos(alpha) ** 2 * s ** 2
+
+    F3 = lambda alpha, phi, s: 1 + (1 + eta(phi) ** 2) / (12 * re.N(phi) ** 2) * sin(alpha) ** 2 * s ** 2 \
+                               + (3 + 8 * eta(phi)**2) / (24 * re.N(phi) ** 2) * cos(alpha) ** 2 * s ** 2
+
+    phi1_i = lambda alpha, phi: phi0 + cos(alpha) / re.M(phi) * s * F1(alpha, phi, s)
+    lamb1_i = lambda alpha, phi: lamb0 + sin(alpha) / (re.N(phi) * cos(phi)) * s * F2(alpha, phi, s)
+    alpha1_i = lambda alpha, phi: alpha0 + sin(alpha) * tan(phi) / re.N(phi) * s * F3(alpha, phi, s)
+
+    phi_m_i = lambda phi1: (phi0 + phi1) / 2
+    alpha_m_i = lambda alpha1: (alpha0 + alpha1) / 2
+
+    phi_m = []
+    alpha_m = []
+    phi1 = []
+    lamb1 = []
+    alpha1 = []
+
+    # 1. Näherung für P1
+    phi1.append(phi0 + cos(alpha0) / re.M(phi0) * s)
+    lamb1.append(lamb0 + sin(alpha0) / (re.N(phi0) * cos(phi0)) * s)
+    alpha1.append(alpha0 + sin(alpha0) * tan(phi0) / re.N(phi0) * s)
+
+    while True:
+        # Berechnug P_m durch Mittelbildung
+        phi_m.append(phi_m_i(phi1[-1]))
+        alpha_m.append(alpha_m_i(alpha1[-1]))
+        # Berechnung P1
+        phi1.append(phi1_i(alpha_m[-1], phi_m[-1]))
+        lamb1.append(lamb1_i(alpha_m[-1], phi_m[-1]))
+        alpha1.append(alpha1_i(alpha_m[-1], phi_m[-1]))
+
+        # Abbruchkriterium
+        if abs(phi1[-2] - phi1[-1]) < eps and \
+           abs(lamb1[-2] - lamb1[-1]) < eps and \
+           abs(alpha1[-2] - alpha1[-1]) < eps:
+            break
+    return phi1[-1], lamb1[-1], alpha1[-1]
+
+
+def gha2(re: EllipsoidBiaxial, phi0: float, lamb0: float, phi1: float, lamb1: float):
+    """
+    Berechnung der 2. Geodätische Hauptaufgabe nach Gauß´schen Mittelbreitenformeln
+    :param re: Klasse Ellipsoid
+    :param phi0: Breite Punkt 1
+    :param lamb0: Länge Punkt 1
+    :param phi1: Breite Punkt 2
+    :param lamb1: Länge Punkt 2
+    :return: Länge der geodätischen Linie, Azimut von P1 nach P2, Azimut von P2 nach P1
+    """
+
+    t = lambda phi: tan(phi)
+    eta = lambda phi: sqrt(re.e_ ** 2 * cos(phi) ** 2)
+
+    phi_0 = (phi0 + phi1) / 2
+    d_phi = phi1 - phi0
+    d_lambda = lamb1 - lamb0
+
+    f_A = lambda phi: (2 + 3*t(phi)**2 + 2*eta(phi)**2) / 24
+    f_B = lambda phi: ((t(phi)**2 - 1) * eta(phi)**2) / 8
+    # f_C = lambda phi: (t(phi)**2) / 24
+    f_D = lambda phi: (1 + eta(phi)**2 - 9 * eta(phi)**2 * t(phi)**2) / 24
+
+    F1 = lambda phi: d_phi * re.M(phi) * (1 - f_A(phi) * d_lambda ** 2 * cos(phi) ** 2 -
+                                          f_B(phi) * d_phi ** 2 / re.V(phi) ** 4)
+    F2 = lambda phi: d_lambda * re.N(phi) * cos(phi) * (1 - 1 / 24 * d_lambda ** 2 * sin(phi) ** 2 +
+                                                           f_D(phi) * d_phi ** 2 / re.V(phi) ** 4)
+
+    s = sqrt(F1(phi_0) ** 2 + F2(phi_0) ** 2)
+    A_0 = arctan(F2(phi_0) / F1(phi_0))
+    d_A = d_lambda * sin(phi_0) * (1 + (1 + eta(phi_0) ** 2) / 12 * s ** 2 * sin(A_0) ** 2 / re.N(phi_0) ** 2 +
+                                      (3 + 8 * eta(phi_0) ** 2) / 24 * s ** 2 * cos(A_0) ** 2 / re.N(phi_0) ** 2)
+
+    alpha0 = A_0 - d_A / 2
+    alpha1 = A_0 + d_A / 2
+
+    return alpha0, alpha1, s

nicht abgeben/GHA_biaxial/rk.py

@@ -0,0 +1,33 @@
+from typing import Tuple
+
+import numpy as np
+from ellipsoid_biaxial import EllipsoidBiaxial
+from numpy import cos, sin, tan
+from numpy.typing import NDArray
+
+import runge_kutta as rk
+
+
+def gha1(re: EllipsoidBiaxial, phi0: float, lamb0: float, alpha0: float, s: float, num: int) -> Tuple[float, float, float]:
+    """
+    Berechnung der 1. GHA auf einem Rotationsellipsoid mittels RK4
+    :param re:
+    :param phi0:
+    :param lamb0:
+    :param alpha0:
+    :param s:
+    :param num:
+    :return:
+    """
+    def buildODE():
+        def ODE(s: float, v: NDArray):
+            phi, lam, A = v
+            V = re.V(phi)
+            dphi = cos(A) * V ** 3 / re.c
+            dlam = sin(A) * V / (cos(phi) * re.c)
+            dA = tan(phi) * sin(A) * V / re.c
+            return np.array([dphi, dlam, dA])
+        return ODE
+
+    _, funktionswerte = rk.rk4(buildODE(), 0, np.array([phi0, lamb0, alpha0]), s, num)
+    return funktionswerte[-1][0], funktionswerte[-1][1], funktionswerte[-1][2]

nicht abgeben/Tests/alpha_conversion_test.ipynb

@@ -0,0 +1,101 @@
+{
+ "cells": [
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "%load_ext autoreload\n",
+    "%autoreload 2"
+   ],
+   "id": "a78faf7f4883772f",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "%reload_ext autoreload\n",
+    "%autoreload 2\n",
+    "import numpy as np\n",
+    "\n",
+    "import winkelumrechnungen as wu\n",
+    "from GHA_triaxial.utils import alpha_ell2para, alpha_para2ell\n",
+    "from ellipsoid_triaxial import EllipsoidTriaxial"
+   ],
+   "id": "46aa84a937fea491",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "ell = EllipsoidTriaxial.init_name(\"KarneyTest2024\")\n",
+    "diffs = []\n",
+    "for beta_deg in range(-90, 91, 15):\n",
+    "    for lamb_deg in range(-180, 180, 15):\n",
+    "        for alpha_deg in range(0, 360, 15):\n",
+    "            beta = wu.deg2rad(beta_deg)\n",
+    "            lamb = wu.deg2rad(lamb_deg)\n",
+    "            u, v = ell.ell2para(beta, lamb)\n",
+    "            alpha = wu.deg2rad(alpha_deg)\n",
+    "\n",
+    "            alpha_para_1, *_ = alpha_ell2para(ell, beta, lamb, alpha)\n",
+    "            alpha_ell_1, *_ = alpha_para2ell(ell, u, v, alpha_para_1)\n",
+    "            diff_1 = wu.deg2rad(abs(alpha_ell_1 - alpha))/3600\n",
+    "\n",
+    "            alpha_ell_2, *_ = alpha_para2ell(ell, u, v, alpha)\n",
+    "            alpha_para_2, *_ = alpha_ell2para(ell, beta, lamb, alpha_ell_2)\n",
+    "            diff_2 = wu.deg2rad(abs(alpha_para_2 - alpha))/3600\n",
+    "\n",
+    "            diffs.append((beta_deg, lamb_deg, alpha_deg, diff_1, diff_2))\n",
+    "diffs = np.array(diffs)"
+   ],
+   "id": "82fc6cbbe7d5abcb",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "i_max_ell = np.argmax(diffs[:, 3])\n",
+    "max_ell = diffs[i_max_ell, 3]\n",
+    "point_max_ell = diffs[i_max_ell, :3]\n",
+    "\n",
+    "i_max_para = np.argmax(diffs[:, 4])\n",
+    "max_para = diffs[i_max_para, 4]\n",
+    "point_max_para = diffs[i_max_para, :4]\n",
+    "\n",
+    "print(f'Für elliptisches Alpha = {point_max_ell[2]}° und beta = {point_max_ell[0]}°, lamb = {point_max_ell[1]}°: diff = {max_ell}\"')\n",
+    "print(f'Für parametrisches Alpha = {point_max_para[2]}° und beta = {point_max_para[0]}°, lamb = {point_max_para[1]}°: diff = {max_ell}\"')\n",
+    "pass"
+   ],
+   "id": "97b5b8c9ca5377ab",
+   "outputs": [],
+   "execution_count": null
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 2
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython2",
+   "version": "2.7.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

nicht abgeben/Tests/conversions_test.ipynb

@@ -0,0 +1,246 @@
+{
+ "cells": [
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "%load_ext autoreload\n",
+    "%autoreload 2"
+   ],
+   "id": "746c5b9e4c0226e7",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "cell_type": "code",
+   "id": "initial_id",
+   "metadata": {
+    "collapsed": true
+   },
+   "source": [
+    "%reload_ext autoreload\n",
+    "%autoreload 2\n",
+    "from itertools import product\n",
+    "\n",
+    "import numpy as np\n",
+    "import pandas as pd\n",
+    "import plotly.graph_objects as go\n",
+    "\n",
+    "import winkelumrechnungen as wu\n",
+    "from ellipsoid_triaxial import EllipsoidTriaxial"
+   ],
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "# ellips = \"KarneyTest2024\"\n",
+    "ellips = \"BursaSima1980\"\n",
+    "# ellips = \"Fiction\"\n",
+    "ell: EllipsoidTriaxial = EllipsoidTriaxial.init_name(ellips)"
+   ],
+   "id": "7b05ca89fcd7b331",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "def deg_range(start, stop, step):\n",
+    "    return [float(x) for x in range(start, stop + step, step)]\n",
+    "\n",
+    "def asymptotic_range(start, direction=\"up\", max_decimals=4):\n",
+    "    values = []\n",
+    "    for d in range(0, max_decimals + 1):\n",
+    "        step = 10 ** -d\n",
+    "        if direction == \"up\":\n",
+    "            values.append(start + (1 - step))\n",
+    "        else:\n",
+    "            values.append(start - (1 - step))\n",
+    "    return values"
+   ],
+   "id": "61a6b14fef0180ad",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "beta_5_85 = deg_range(5, 85, 5)\n",
+    "lambda_5_85 = deg_range(5, 85, 5)\n",
+    "beta_5_90 = deg_range(5, 90, 5)\n",
+    "lambda_5_90 = deg_range(5, 90, 5)\n",
+    "beta_0_90 = deg_range(0, 90, 5)\n",
+    "lambda_0_90 = deg_range(0, 90, 5)\n",
+    "beta_90 = [90.0]\n",
+    "lambda_90 = [90.0]\n",
+    "beta_0 = [0.0]\n",
+    "lambda_0 = [0.0]\n",
+    "beta_asym_89 = asymptotic_range(89.0, direction=\"up\")\n",
+    "lambda_asym_0 = asymptotic_range(1.0, direction=\"down\")"
+   ],
+   "id": "f7184980a4b930b7",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "groups = {\n",
+    "    1: list(product(beta_5_85, lambda_5_85)),\n",
+    "    2: list(product(beta_0, lambda_0_90)),\n",
+    "    3: list(product(beta_5_85, lambda_0)),\n",
+    "    4: list(product(beta_90, lambda_5_90)),\n",
+    "    5: list(product(beta_asym_89, lambda_asym_0)),\n",
+    "    6: list(product(beta_5_85, lambda_90)),\n",
+    "    7: list(product(lambda_asym_0, lambda_0_90)),\n",
+    "    8: list(product(beta_0_90, lambda_asym_0)),\n",
+    "    9: list(product(beta_asym_89, lambda_0_90)),\n",
+    "    10: list(product(beta_0_90, beta_asym_89)),\n",
+    "}"
+   ],
+   "id": "cea9fd9cce6a4fd1",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "for nr, points in groups.items():\n",
+    "    points_cart = []\n",
+    "    for point in points:\n",
+    "        beta, lamb = point\n",
+    "        cart = ell.ell2cart(wu.deg2rad(beta), wu.deg2rad(lamb))\n",
+    "        points_cart.append(cart)\n",
+    "    groups[nr] = points_cart"
+   ],
+   "id": "17a6a130782a89ce",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "results = {}\n",
+    "\n",
+    "for nr, points in groups.items():\n",
+    "    group_results = {\"ell\": [],\n",
+    "                     \"para\": [],\n",
+    "                     \"geod\": []}\n",
+    "    for point in points:\n",
+    "        elli = ell.cart2ell(point)\n",
+    "        cart_elli = ell.ell2cart(elli[0], elli[1])\n",
+    "        group_results[\"ell\"].append(np.linalg.norm(point - cart_elli, axis=-1))\n",
+    "\n",
+    "        para = ell.cart2para(point)\n",
+    "        cart_para = ell.para2cart(para[0], para[1])\n",
+    "        group_results[\"para\"].append(np.linalg.norm(point - cart_para, axis=-1))\n",
+    "\n",
+    "        geod = ell.cart2geod(point, \"ligas3\")\n",
+    "        cart_geod = ell.geod2cart(geod[0], geod[1], geod[2])\n",
+    "        group_results[\"geod\"].append(np.linalg.norm(point - cart_geod, axis=-1))\n",
+    "\n",
+    "    group_results[\"ell\"] = np.array(group_results[\"ell\"])\n",
+    "    group_results[\"para\"] = np.array(group_results[\"para\"])\n",
+    "    group_results[\"geod\"] = np.array(group_results[\"geod\"])\n",
+    "    results[nr] = group_results"
+   ],
+   "id": "c3298ea233bca274",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "# with open(f\"conversion_results_{ellips}.pkl\", \"wb\") as f:\n",
+    "#     pickle.dump(results, f)"
+   ],
+   "id": "e1285860be416ad3",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "# with open(f\"conversion_results_{ellips}.pkl\", \"rb\") as f:\n",
+    "#     results = pickle.load(f)"
+   ],
+   "id": "d26720e34595ccbc",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "df = pd.DataFrame({\n",
+    "    \"Gruppe\": [nr for nr in results.keys()],\n",
+    "    \"max_Δr_ell\": [f\"{max(result[\"ell\"]):.3g}\" for result in results.values()],\n",
+    "    \"max_Δr_para\": [f\"{max(result[\"para\"]):.3g}\" for result in results.values()],\n",
+    "    \"max_Δr_geod\": [f\"{max(result[\"geod\"]):.3g}\" for result in results.values()]\n",
+    "})"
+   ],
+   "id": "4e2e55e4699ec81e",
+   "outputs": [],
+   "execution_count": null
+  },
+  {
+   "metadata": {},
+   "cell_type": "code",
+   "source": [
+    "fig = go.Figure(data=[go.Table(\n",
+    "    header=dict(\n",
+    "        values=list(df.columns),\n",
+    "        fill_color=\"lightgrey\",\n",
+    "        align=\"left\"\n",
+    "    ),\n",
+    "    cells=dict(\n",
+    "        values=[df[col] for col in df.columns],\n",
+    "        align=\"left\"\n",
+    "    )\n",
+    ")])\n",
+    "fig.update_layout(\n",
+    "    template=\"simple_white\",\n",
+    "    width=650,\n",
+    "    height=len(groups)*20+80,\n",
+    "    margin=dict(l=20, r=20, t=20, b=20))\n",
+    "\n",
+    "fig.show()\n",
+    "# fig.write_image(f\"conversion_results_{ellips}.png\", width=650, height=len(groups)*20+80, scale=2)"
+   ],
+   "id": "c2fa82afef2d6e0e",
+   "outputs": [],
+   "execution_count": null
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "Python 3",
+   "language": "python",
+   "name": "python3"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 2
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython2",
+   "version": "2.7.6"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}

nicht abgeben/ellipsoid_biaxial.py

@@ -0,0 +1,126 @@
+from typing import Tuple
+
+import numpy as np
+from numpy import arctan2, cos, sin, sqrt
+from numpy.typing import NDArray
+
+import winkelumrechnungen as wu
+
+
+class EllipsoidBiaxial:
+    """
+    Klasse für Rotationsellipdoide
+    """
+    def __init__(self, a: float, b: float):
+        self.a = a
+        self.b = b
+        self.c = a ** 2 / b
+        self.e = sqrt(a ** 2 - b ** 2) / a
+        self.e_ = sqrt(a ** 2 - b ** 2) / b
+
+    @classmethod
+    def init_name(cls, name: str) -> EllipsoidBiaxial:
+        """
+        Erstellen eines Rotationsellipdoids nach Namen
+        :param name: Name des Rotationsellipsoids
+        :return: Rotationsellipsoid
+        """
+        if name == "Bessel":
+            a = 6377397.15508
+            b = 6356078.96290
+            return cls(a, b)
+        elif name == "Hayford":
+            a = 6378388
+            f = 1/297
+            b = a - a * f
+            return cls(a, b)
+        elif name == "Krassowski":
+            a = 6378245
+            f = 298.3
+            b = a - a * f
+            return cls(a, b)
+        elif name == "WGS84":
+            a = 6378137
+            f = 298.257223563
+            b = a - a * f
+            return cls(a, b)
+        else:
+            raise Exception(f"EllipsoidBiaxial.init_name: Name {name} unbekannt")
+
+    @classmethod
+    def init_af(cls, a: float, f: float) -> EllipsoidBiaxial:
+        """
+        Erstellen eines Rotationsellipdoids aus der großen Halbachse und der Abplattung
+        :param a: große Halbachse
+        :param f: großen Halbachse
+        :return: Rotationsellipsoid
+        """
+        b = a - a * f
+        return cls(a, b)
+
+    V = lambda self, phi: sqrt(1 + self.e_ ** 2 * cos(phi) ** 2)
+    M = lambda self, phi: self.c / self.V(phi) ** 3
+    N = lambda self, phi: self.c / self.V(phi)
+
+    beta2psi = lambda self, beta: arctan2(self.a * sin(beta), self.b * cos(beta))
+    beta2phi = lambda self, beta: arctan2(self.a ** 2 * sin(beta), self.b ** 2 * cos(beta))
+
+    psi2beta = lambda self, psi: arctan2(self.b * sin(psi), self.a * cos(psi))
+    psi2phi = lambda self, psi: arctan2(self.a * sin(psi), self.b * cos(psi))
+
+    phi2beta = lambda self, phi: arctan2(self.b**2 * sin(phi), self.a**2 * cos(phi))
+    phi2psi = lambda self, phi: arctan2(self.b * sin(phi), self.a * cos(phi))
+
+    phi2p = lambda self, phi: self.N(phi) * cos(phi)
+
+    def bi_cart2ell(self, point: NDArray, Eh: float = 0.001, Ephi: float = wu.gms2rad([0, 0, 0.001])) -> Tuple[float, float, float]:
+        """
+        Umrechnung von kartesischen in ellipsoidische Koordinaten auf einem Rotationsellipsoid
+        # TODO: Quelle
+        :param point: Punkt in kartesischen Koordinaten
+        :param Eh: Grenzwert für die Höhe
+        :param Ephi: Grenzwert für die Breite
+        :return: ellipsoidische Breite, Länge, geodätische Höhe
+        """
+        x, y, z = point
+
+        lamb = arctan2(y, x)
+
+        p = sqrt(x**2+y**2)
+
+        phi_null = arctan2(z, p*(1 - self.e**2))
+
+        hi = [0]
+        phii = [phi_null]
+
+        i = 0
+
+        while True:
+            N = self.a / sqrt(1 - self.e**2 * sin(phii[i])**2)
+            h = p / cos(phii[i]) - N
+            phi = arctan2(z,  p * (1-(self.e**2*N) / (N+h)))
+            hi.append(h)
+            phii.append(phi)
+            dh = abs(hi[i]-h)
+            dphi = abs(phii[i]-phi)
+            i += 1
+            if dh < Eh:
+                if dphi < Ephi:
+                    break
+        return phi, lamb, h
+
+    def bi_ell2cart(self, phi: float, lamb: float, h: float) -> NDArray:
+        """
+        Umrechnung von ellipsoidischen in kartesische Koordinaten auf einem Rotationsellipsoid
+        # TODO: Quelle
+        :param phi: ellipsoidische Breite
+        :param lamb: ellipsoidische Länge
+        :param h: geodätische Höhe
+        :return: Punkt in kartesischen Koordinaten
+        """
+        W = sqrt(1 - self.e**2 * sin(phi)**2)
+        N = self.a / W
+        x = (N+h) * cos(phi) * cos(lamb)
+        y = (N+h) * cos(phi) * sin(lamb)
+        z = (N * (1-self.e**2) + h) * sin(phi)
+        return np.array([x, y, z])

nicht abgeben/kugel.py

@@ -0,0 +1,100 @@
+from typing import Tuple
+
+import numpy as np
+from numpy import arccos, arcsin, arctan2, cos, pi, sin, sqrt
+from numpy.typing import NDArray
+
+import winkelumrechnungen as wu
+
+
+def cart2sph(point: NDArray) -> Tuple[float, float, float]:
+    """
+    Umrechnung von kartesischen in sphärische Koordinaten
+    # TODO: Quelle
+    :param point: Punkt in kartesischen Koordinaten
+    :return: Radius, Breite, Länge
+    """
+    x, y, z = point
+    r = sqrt(x**2 + y**2 + z**2)
+    phi = arctan2(z, sqrt(x**2 + y**2))
+    lamb = arctan2(y, x)
+
+    return r, phi, lamb
+
+
+def sph2cart(r: float, phi: float, lamb: float) -> NDArray:
+    """
+    Umrechnung von sphärischen in kartesische Koordinaten
+    # TODO: Quelle
+    :param r: Radius
+    :param phi: Breite
+    :param lamb: Länge
+    :return: Punkt in kartesischen Koordinaten
+    """
+    x = r * cos(phi) * cos(lamb)
+    y = r * cos(phi) * sin(lamb)
+    z = r * sin(phi)
+
+    return np.array([x, y, z])
+
+
+def gha1(R: float, phi0: float, lamb0: float, s: float, alpha0: float) -> Tuple[float, float]:
+    """
+    Berechnung der 1. GHA auf der Kugel
+    # TODO: Quelle
+    :param R: Radius
+    :param phi0: Breite des Startpunktes
+    :param lamb0: Länge des Startpunktes
+    :param s: Strecke
+    :param alpha0: Azimut
+    :return: Breite, Länge des Zielpunktes
+    """
+    s_ = s / R
+
+    lamb1 = lamb0 + arctan2(sin(s_) * sin(alpha0),
+                            cos(phi0) * cos(s_) - sin(phi0) * sin(s_) * cos(alpha0))
+
+    phi1 = arcsin(sin(phi0) * cos(s_) + cos(phi0) * sin(s_) * cos(alpha0))
+
+    return phi1, lamb1
+
+
+def gha2(R: float, phi0: float, lamb0: float, phi1: float, lamb1: float) -> Tuple[float, float, float]:
+    """
+    Berechnung der 2. GHA auf der Kugel
+    # TODO: Quelle
+    :param R: Radius
+    :param phi0: Breite des Startpunktes
+    :param lamb0: Länge des Startpunktes
+    :param phi1: Breite des Zielpunktes
+    :param lamb1: Länge des Zielpunktes
+    :return: Azimut im Startpunkt, Azimut im Zielpunkt, Strecke
+    """
+    s_ = arccos(sin(phi0) * sin(phi1) + cos(phi0) * cos(phi1) * cos(lamb1 - lamb0))
+    s = R * s_
+
+    alpha0 = arctan2(cos(phi1) * sin(lamb1 - lamb0),
+                     cos(phi0) * sin(phi1) - sin(phi0) * cos(phi1) * cos(lamb1 - lamb0))
+
+    alpha1 = arctan2(-cos(phi0) * sin(lamb1 - lamb0),
+                     cos(phi1) * sin(phi0) - sin(phi1) * cos(phi0) * cos(lamb1 - lamb0))
+    if alpha1 < 0:
+        alpha1 += 2 * pi
+
+    return alpha0, alpha1, s
+
+
+if __name__ == "__main__":
+    R = 6378815.904                                             # Bern
+    phi0 = wu.deg2rad(10)
+    lamb0 = wu.deg2rad(40)
+    alpha0 = wu.deg2rad(100)
+    s = 10000
+    phi1, lamb1 = gha1(R, phi0, lamb0, s, alpha0)
+    alpha0_g, alpha1, s_g = gha2(R, phi0, lamb0, phi1, lamb1)
+    phi1 = wu.rad2deg(phi1)
+    lamb1 = wu.rad2deg(lamb1)
+    alpha0_g = wu.rad2deg(alpha0_g)
+    alpha1 = wu.rad2deg(alpha1)
+
+    pass

nicht abgeben/test.py

@@ -0,0 +1,8 @@
+import numpy as np
+
+import ellipsoid_triaxial
+
+ell = ellipsoid_triaxial.EllipsoidTriaxial.init_name("KarneyTest2024")
+
+cart = ell.para2cart(0, np.pi/2)
+print(cart)

requirements.txt

@@ -0,0 +1,7 @@
+numpy~=2.3.4
+plotly~=6.4.0
+pandas~=2.3.3
+scipy~=1.16.3
+dash-bootstrap-components~=2.0.4
+dash~=4.0.0
+matplotlib~=3.10.7

runge_kutta.py

@@ -1,7 +1,10 @@
+from typing import Callable
+
 import numpy as np
+from numpy.typing import NDArray
 
 
-def rk4(ode, t0: float, v0: np.ndarray, weite: float, schritte: int, fein: bool = False) -> tuple[list, list]:
+def rk4(ode: Callable, t0: float, v0: NDArray, weite: float, schritte: int, fein: bool = False) -> tuple[list, list]:
     """
     Standard Runge-Kutta Verfahren 4. Ordnung
     :param ode: ODE-System als Funktion
@@ -9,7 +12,7 @@ def rk4(ode, t0: float, v0: np.ndarray, weite: float, schritte: int, fein: bool
     :param v0: Startwerte
     :param weite: Integrationsweite
     :param schritte: Schrittzahl
-    :param fein:
+    :param fein: Fein-Rechnung?
     :return: Variable und Funktionswerte an jedem Stützpunkt
     """
     h = weite/schritte
@@ -35,9 +38,118 @@ def rk4(ode, t0: float, v0: np.ndarray, weite: float, schritte: int, fein: bool
 
     return t_list, werte
 
-def rk4_step(ode, t: float, v: np.ndarray, h: float) -> np.ndarray:
+def rk4_step(ode: Callable, t: float, v: NDArray, h: float) -> NDArray:
+    """
+    Ein Schritt des Runge-Kutta Verfahrens 4. Ordnung
+    :param ode: ODE-System als Funktion
+    :param t: unabhängige Variable
+    :param v: abhängige Variablen
+    :param h: Schrittweite
+    :return: abhängige Variablen nach einem Schritt
+    """
     k1 = ode(t, v)
     k2 = ode(t + 0.5 * h, v + 0.5 * h * k1)
     k3 = ode(t + 0.5 * h, v + 0.5 * h * k2)
     k4 = ode(t + h, v + h * k3)
     return v + (h / 6.0) * (k1 + 2 * k2 + 2 * k3 + k4)
+
+def rk4_end(ode: Callable, t0: float, v0: NDArray, weite: float, schritte: int, fein: bool = False):
+    """
+    Standard Runge-Kutta Verfahren 4. Ordnung, nur Ausgabe der letzten Variablenwerte
+    :param ode: ODE-System als Funktion
+    :param t0: Startwert der unabhängigen Variable
+    :param v0: Startwerte
+    :param weite: Integrationsweite
+    :param schritte: Schrittzahl
+    :param fein: Fein-Rechnung?
+    :return: Variable und Funktionswerte am letzten Stützpunkt
+    """
+    h = weite / schritte
+    t = float(t0)
+    v = np.array(v0, dtype=float, copy=True)
+
+    for _ in range(schritte):
+        if not fein:
+            v_next = rk4_step(ode, t, v, h)
+        else:
+            v_grob = rk4_step(ode, t, v, h)
+            v_half = rk4_step(ode, t, v, 0.5 * h)
+            v_fein = rk4_step(ode, t + 0.5 * h, v_half, 0.5 * h)
+            v_next = v_fein + (v_fein - v_grob) / 15.0
+
+        t += h
+        v = v_next
+
+    return t, v
+
+# RK4 mit Simpson bzw. Trapez
+def rk4_integral(ode: Callable, t0: float, v0: NDArray, weite: float, schritte: int, integrand_at: Callable, fein: bool = False, simpson: bool = True):
+    """
+    Runge-Kutta Verfahren 4. Ordnung mit Simpson bzw. Trapez
+    :param ode: ODE-System als Funktion
+    :param t0: Startwert der unabhängigen Variable
+    :param v0: Startwerte
+    :param weite: Integrationsweite
+    :param integrand_at: Funktion
+    :param schritte: Schrittzahl
+    :param fein: Fein-Rechnung?
+    :param simpson: Simpson? Wenn nein, dann Trapez
+    :return: Variable und Funktionswerte am letzten Stützpunkt
+    """
+    h = weite / schritte
+    habs = abs(h)
+
+    t = float(t0)
+    v = np.array(v0, dtype=float, copy=True)
+
+    if simpson and (schritte % 2 == 0):
+        f0 = float(integrand_at(t, v))
+        odd_sum = 0.0
+        even_sum = 0.0
+        fN = None
+
+        for i in range(1, schritte + 1):
+            if not fein:
+                v_next = rk4_step(ode, t, v, h)
+            else:
+                v_grob = rk4_step(ode, t, v, h)
+                v_half = rk4_step(ode, t, v, 0.5 * h)
+                v_fein = rk4_step(ode, t + 0.5 * h, v_half, 0.5 * h)
+                v_next = v_fein + (v_fein - v_grob) / 15.0
+
+            t += h
+            v = v_next
+
+            fi = float(integrand_at(t, v))
+            if i == schritte:
+                fN = fi
+            elif i % 2 == 1:
+                odd_sum += fi
+            else:
+                even_sum += fi
+
+        S = f0 + fN + 4.0 * odd_sum + 2.0 * even_sum
+        s = (habs / 3.0) * S
+        return t, v, s
+
+    f_prev = float(integrand_at(t, v))
+    acc = 0.0
+
+    for _ in range(schritte):
+        if not fein:
+            v_next = rk4_step(ode, t, v, h)
+        else:
+            v_grob = rk4_step(ode, t, v, h)
+            v_half = rk4_step(ode, t, v, 0.5 * h)
+            v_fein = rk4_step(ode, t + 0.5 * h, v_half, 0.5 * h)
+            v_next = v_fein + (v_fein - v_grob) / 15.0
+
+        t += h
+        v = v_next
+
+        f_cur = float(integrand_at(t, v))
+        acc += 0.5 * (f_prev + f_cur)
+        f_prev = f_cur
+
+    s = habs * acc
+    return t, v, s

test.py

@@ -1,14 +0,0 @@
-import numpy as np
-from scipy.special import factorial as fact
-from math import comb
-import matplotlib.pyplot as plt
-import ellipsoide
-from GHA_triaxial.panou import pq
-
-ell = ellipsoide.EllipsoidTriaxial.init_name("Eitschberger1978")
-
-x, y, z = ell.para2cart(0.5, 0.7)
-
-x1 = pq(ell, x, y, z)
-x2 = ell.p_q(x, y, z)
-pass

utils.py

@@ -1,21 +0,0 @@
-from numpy import arctan2
-from numpy.typing import NDArray
-
-from GHA_triaxial.panou import pq_ell
-from ellipsoide import EllipsoidTriaxial
-
-
-def sigma2alpha(ell: EllipsoidTriaxial, sigma: NDArray, point: NDArray) -> float:
-    """
-    Berechnung des Richtungswinkels an einem Punkt anhand der Ableitung zu den kartesischen Koordinaten
-    :param ell: Ellipsoid
-    :param sigma: Ableitungsvektor ver kartesischen Koordinaten
-    :param point: Punkt
-    :return: Richtungswinkel
-    """
-    p, q = pq_ell(ell, point)
-    P = float(p @ sigma)
-    Q = float(q @ sigma)
-
-    alpha = arctan2(P, Q)
-    return alpha

utils_angle.py

@@ -0,0 +1,54 @@
+import numpy as np
+
+import winkelumrechnungen as wu
+
+
+def arccot(x: float) -> float:
+    """
+    Berechnung von arccot eines Winkels
+    :param x: Winkel
+    :return: arccot(Winkel)
+    """
+    return np.arctan2(1.0, x)
+
+
+def cot(x: float) -> float:
+    """
+    Berechnung von cot eines Winkels
+    :param x: Winkel
+    :return: cot(Winkel)
+    """
+    return np.cos(x) / np.sin(x)
+
+
+def wrap_mpi_pi(x: float) -> float:
+    """
+    Wrap eines Winkels in den Wertebereich [-π, π)
+    :param x: Winkel
+    :return: Winkel in [-π, π)
+    """
+    return (x + np.pi) % (2 * np.pi) - np.pi
+
+
+def wrap_mhalfpi_halfpi(x: float) -> float:
+    """
+    Wrap eines Winkels in den Wertebereich [-π/2, π/2)
+    :param x: Winkel
+    :return: Winkel in [-π/2, π/2)
+    """
+    return (x + np.pi / 2) % np.pi - np.pi / 2
+
+
+def wrap_0_2pi(x: float) -> float:
+    """
+    Wrap eines Winkels in den Wertebereich [0, 2π)
+    :param x: Winkel
+    :return: Winkel in [0, 2π)
+    """
+    return x % (2 * np.pi)
+
+
+if __name__ == "__main__":
+    print(wu.rad2deg(wrap_mhalfpi_halfpi(wu.deg2rad(181))))
+    print(wu.rad2deg(wrap_0_2pi(wu.deg2rad(181))))
+    print(wu.rad2deg(wrap_mpi_pi(wu.deg2rad(181))))