Ellipsoid-Auswahl angepasst

revert Delete Tests/gha_resultsKarney.pkl

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")
@@ -12,7 +13,6 @@ def felli(x):
 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 = {}
 
@@ -91,8 +91,7 @@ def escma(func, *, N=10, xmean=None, sigma=0.5, stopfitness=1e-14, stopeval=2000
             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}')
@@ -106,13 +105,10 @@ def escma(func, *, N=10, xmean=None, sigma=0.5, stopfitness=1e-14, stopeval=2000
             # 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 < \
+        hsig = norm_ps / np.sqrt(1 - (1 - cs) ** (2 * counteval / lambda_)) / chiN < (1.4 + 2 / (N + 1))
-               (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)

ES/gha1_ES.py

@@ -1,29 +1,17 @@
 from __future__ import annotations
 
-from codeop import PyCF_ALLOW_INCOMPLETE_INPUT
+from typing import List, Tuple
-from typing import List, Optional, Tuple
+
 import numpy as np
-from ellipsoide import EllipsoidTriaxial
+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 Hansen_ES_CMA import escma
+from GHA_triaxial.utils import jacobi_konstante
-from utils_angle import wrap_to_pi
+from ellipsoid_triaxial import EllipsoidTriaxial
-from numpy.typing import NDArray
+from utils_angle import wrap_mpi_pi
-import winkelumrechnungen as wu
-
-
-def ellipsoid_formparameter(ell: EllipsoidTriaxial):
-    """
-    Berechnet die Formparameter des dreiachsigen Ellipsoiden nach Karney (2025), Gl. (2)
-    :param ell: Ellipsoid
-    :return: e, k und k'
-    """
-    nenner = np.sqrt(max(ell.ax * ell.ax - ell.b * ell.b, 0.0))
-    k = np.sqrt(max(ell.ay * ell.ay - ell.b * ell.b, 0.0)) / nenner
-    k_ = np.sqrt(max(ell.ax * ell.ax - ell.ay * ell.ay, 0.0)) / nenner
-    e = np.sqrt(max(ell.ax * ell.ax - ell.b * ell.b, 0.0)) / ell.ay
-
-    return e, k, k_
 
 
 def ENU_beta_omega(beta: float, omega: float, ell: EllipsoidTriaxial) \
@@ -40,7 +28,7 @@ def ENU_beta_omega(beta: float, omega: float, ell: EllipsoidTriaxial) \
                 R (XYZ) = Punkt in XYZ
     """
     # Berechnungshilfen
-    omega = wrap_to_pi(omega)
+    omega = wrap_mpi_pi(omega)
     cb = np.cos(beta)
     sb = np.sin(beta)
     co = np.cos(omega)
@@ -74,34 +62,19 @@ def ENU_beta_omega(beta: float, omega: float, ell: EllipsoidTriaxial) \
     # 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 = np.linalg.norm(E)
+    En = float(np.linalg.norm(E))
-    Nn = np.linalg.norm(N)
+    Nn = float(np.linalg.norm(N))
-    Un = np.linalg.norm(U)
+    Un = float(np.linalg.norm(U))
 
     N_hat = N / Nn
     E_hat = E / En
     U_hat = U / Un
-    E_hat = E_hat - float(np.dot(E_hat, N_hat)) * N_hat
+    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 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
-    """
-    e, k, k_ = ellipsoid_formparameter(ell)
-    gamma_jacobi = float((k ** 2) * (np.cos(beta) ** 2) * (np.sin(alpha) ** 2) - (k_ ** 2) * (np.sin(omega) ** 2) * (np.cos(alpha) ** 2))
-
-    return gamma_jacobi
-
-
 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
@@ -117,11 +90,10 @@ def azimuth_at_ESpoint(P_prev: NDArray, P_curr: NDArray, E_hat_curr: NDArray, N_
     vT = v - float(np.dot(v, U_hat_curr)) * U_hat_curr
     vTn = max(np.linalg.norm(vT), 1e-18)
     vT_hat = vT / vTn
-    #vT_hat = vT / np.linalg.norm(vT)
     sE = float(np.dot(vT_hat, E_hat_curr))
     sN = float(np.dot(vT_hat, N_hat_curr))
 
-    return wrap_to_pi(float(np.arctan2(sE, sN)))
+    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,
@@ -154,11 +126,10 @@ def optimize_next_point(beta_i: float, omega_i: float, alpha_i: float, ds: float
     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_to_pi(omega_i + d_omega)
+    omega_pred = wrap_mpi_pi(omega_i + d_omega)
 
     xmean = np.array([beta_pred, omega_pred], dtype=float)
 
@@ -175,7 +146,7 @@ def optimize_next_point(beta_i: float, omega_i: float, alpha_i: float, ds: float
         :return: Fitnesswert (f)
         """
         beta = x[0]
-        omega = wrap_to_pi(x[1])
+        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
@@ -197,11 +168,10 @@ def optimize_next_point(beta_i: float, omega_i: float, alpha_i: float, ds: float
 
         return f
 
-
     xb = escma(fitness, N=2, xmean=xmean, sigma=sigma0)  # Aufruf CMA-ES
 
     beta_best = xb[0]
-    omega_best = wrap_to_pi(xb[1])
+    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)
@@ -209,7 +179,7 @@ def optimize_next_point(beta_i: float, omega_i: float, alpha_i: float, ds: float
     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: boolean = False)\
+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
@@ -223,8 +193,8 @@ def gha1_ES(ell: EllipsoidTriaxial, beta0: float, omega0: float, alpha0: float,
     :return: Zielpunkt Pk, Azimut am Zielpunkt und Punktliste
     """
     beta = float(beta0)
-    omega = wrap_to_pi(float(omega0))
+    omega = wrap_mpi_pi(float(omega0))
-    alpha = wrap_to_pi(float(alpha0))
+    alpha = wrap_mpi_pi(float(alpha0))
 
     gamma0 = jacobi_konstante(beta, omega, alpha, ell)  # Referenz-γ0
 
@@ -243,9 +213,9 @@ def gha1_ES(ell: EllipsoidTriaxial, beta0: float, omega0: float, alpha0: float,
                                                     ell=ell, maxSegLen=maxSegLen)
         s_acc += ds
         P_all.append(P)
-        alpha_end.append(alpha)
+        alpha_end.append(wrap_mpi_pi(alpha))
         if step > nsteps_est + 50:
-            raise RuntimeError("Zu viele Schritte – vermutlich Konvergenzproblem / falsche Azimut-Konvention.")
+            raise RuntimeError("GHA1_ES: Zu viele Schritte – vermutlich Konvergenzproblem / falsche Azimut-Konvention.")
     Pk = P_all[-1]
     alpha1 = float(alpha_end[-1])
 

ES/gha2_ES.py

@@ -1,14 +1,13 @@
-import numpy as np
-from Hansen_ES_CMA import escma
-from ellipsoide import EllipsoidTriaxial
-from numpy.typing import NDArray
 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, pq_ell
+from GHA_triaxial.utils import sigma2alpha
-
+from ellipsoid_triaxial import EllipsoidTriaxial
-
-
 
 
 def Sehne(P1: NDArray, P2: NDArray) -> float:
@@ -19,14 +18,11 @@ def Sehne(P1: NDArray, P2: NDArray) -> float:
     :return: Bogenlänge s
     """
     R12 = P2-P1
-    s = np.linalg.norm(R12)
+    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.
@@ -109,7 +105,7 @@ def gha2_ES(ell: EllipsoidTriaxial, P0: NDArray, Pk: NDArray, maxSegLen: float =
                 startIter += 1
                 maxIter = 10000
                 if startIter > maxIter:
-                    raise RuntimeError("Abbruch: maximale Iterationen überschritten.")
+                    raise RuntimeError("GHA2_ES: maximale Iterationen überschritten")
             new_points.append(B)
 
         points = new_points

GHA_triaxial/gha1_ana.py

@@ -3,12 +3,13 @@ from math import comb
 from typing import Tuple
 
 import numpy as np
-from numpy import sin, cos, arctan2
+from numpy import arctan2, cos, sin
-from numpy._typing import NDArray
+from numpy.typing import NDArray
-import winkelumrechnungen as wu
 
-from ellipsoide import EllipsoidTriaxial
+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]:
@@ -110,7 +111,7 @@ def gha1_ana_step(ell: EllipsoidTriaxial, point: NDArray, alpha0: float, s: floa
     if alpha1 < 0:
         alpha1 += 2 * np.pi
 
-    return p1, alpha1
+    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]:
@@ -132,9 +133,9 @@ def gha1_ana(ell: EllipsoidTriaxial, point: NDArray, alpha0: float, s: float, ma
 
     _, _, h = ell.cart2geod(point_end, "ligas3")
     if h > 1e-5:
-        raise Exception("Analytische Methode ist explodiert, Punkt liegt nicht mehr auf dem Ellipsoid")
+        raise Exception("GHA1_ana: explodiert, Punkt liegt nicht mehr auf dem Ellipsoid")
 
-    return point_end, alpha_end
+    return point_end, wrap_0_2pi(alpha_end)
 
 
 if __name__ == "__main__":
@@ -142,4 +143,3 @@ if __name__ == "__main__":
     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

@@ -1,11 +1,18 @@
-import numpy as np
+from typing import Tuple
-from ellipsoide import EllipsoidTriaxial
-from GHA_triaxial.gha1_ana import gha1_ana
-from GHA_triaxial.utils import func_sigma_ell, louville_constant
-import plotly.graph_objects as go
-import winkelumrechnungen as wu
 
-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
@@ -20,6 +27,8 @@ 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)
@@ -29,21 +38,32 @@ def gha1_approx(ell: EllipsoidTriaxial, p0: np.ndarray, alpha0: float, s: float,
         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)
 
-        dalpha = 1e-6
+        dalpha = 1e-9
         l2 = louville_constant(ell, p2, alpha1)
         dl_dalpha = (louville_constant(ell, p2, alpha1+dalpha) - l2) / dalpha
+        if abs(dl_dalpha) < 1e-20:
+            alpha2 = alpha1 + 0
+        else:
             alpha2 = alpha1 + (l0 - l2) / dl_dalpha
-
         points.append(p2)
-        alphas.append(alpha2)
+        alphas.append(wrap_0_2pi(alpha2))
 
         ds_step = np.linalg.norm(p2 - p1)
         s_curr += ds_step
-        if s_curr > 10000000:
+        last_sigma = sigma
         pass
 
     if all_points:
@@ -78,11 +98,11 @@ def show_points(points: NDArray, p0: NDArray, p1: NDArray):
 
 
 if __name__ == '__main__':
-    ell = EllipsoidTriaxial.init_name("BursaSima1980round")
+    ell = EllipsoidTriaxial.init_name("KarneyTest2024")
-    P0 = ell.para2cart(0.2, 0.3)
+    P0 = ell.ell2cart(wu.deg2rad(15), wu.deg2rad(15))
-    alpha0 = wu.deg2rad(35)
+    alpha0 = wu.deg2rad(270)
-    s = 13000000
+    s = 1
-    P1_app, alpha1_app, points, alphas = gha1_approx(ell, P0, alpha0, s, ds=10000, all_points=True)
+    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=60, maxPartCircum=16)
+    # P1_ana, alpha1_ana = gha1_ana(ell, P0, alpha0, s, maxM=40, maxPartCircum=32)
-    show_points(points, P0, P1_ana)
+    # print(np.linalg.norm(P1_app - P1_ana))
-    print(np.linalg.norm(P1_app - P1_ana))
+    # show_points(points, P0, P0)

GHA_triaxial/gha1_num.py

@@ -1,15 +1,16 @@
+from typing import Callable, List, Tuple
+
 import numpy as np
-from numpy import sin, cos, arctan2
+from numpy import arctan2, cos, sin
-import ellipsoide
-import runge_kutta as rk
-import winkelumrechnungen as wu
-import GHA_triaxial.numeric_examples_karney as ne_karney
-from GHA_triaxial.gha1_ana import gha1_ana
-from ellipsoide import EllipsoidTriaxial
-from typing import Callable, Tuple, List
 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:
@@ -75,8 +76,11 @@ def gha1_num(ell: EllipsoidTriaxial, point: NDArray, alpha0: float, s: float, nu
 
     alpha1 = arctan2(P, Q)
 
-    if alpha1 < 0:
+    alpha1 = wrap_0_2pi(alpha1)
-        alpha1 += 2 * np.pi
+
+    _, _, 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
@@ -104,7 +108,7 @@ if __name__ == "__main__":
     # diffs_panou[mask_360] = np.abs(diffs_panou[mask_360] - 360)
     # print(diffs_panou)
 
-    ell: EllipsoidTriaxial = ellipsoide.EllipsoidTriaxial.init_name("KarneyTest2024")
+    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)

GHA_triaxial/gha2_approx.py

@@ -1,12 +1,13 @@
-import numpy as np
-from ellipsoide import EllipsoidTriaxial
-from GHA_triaxial.gha2_num import gha2_num
-import plotly.graph_objects as go
-import winkelumrechnungen as wu
-from numpy.typing import NDArray
 from typing import Tuple
 
+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_approx(ell: EllipsoidTriaxial, p0: NDArray, p1: NDArray, ds: float, all_points: bool = False) -> Tuple[float, float, float] | Tuple[float, float, float, NDArray]:

GHA_triaxial/gha2_num.py

@@ -1,22 +1,30 @@
+from typing import Tuple
+
 import numpy as np
-from ellipsoide import EllipsoidTriaxial
+from numpy.typing import NDArray
-from runge_kutta import rk4, rk4_step, rk4_end, rk4_integral
+
 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
 import ausgaben as aus
-from utils_angle import cot, arccot, wrap_to_pi
+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):
+def norm_a(a: float) -> float:
-    if a < 0.0:
+    a = float(a) % (2 * np.pi)
-        a += 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_to_pi(lam2 - lam1)
+    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)
@@ -24,6 +32,7 @@ def sph_azimuth(beta1, lam1, beta2, lam2):
         a += 2 * np.pi
     return a
 
+
 # Panou 2013
 def gha2_num(
     ell: EllipsoidTriaxial,
@@ -49,65 +58,67 @@ def gha2_num(
     :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):
-        BETA = (ell.ay ** 2 * np.sin(beta) ** 2 + ell.b ** 2 * np.cos(beta) ** 2) / (
+        sb = np.sin(beta)
-            ell.Ex ** 2 - ell.Ey ** 2 * np.sin(beta) ** 2
+        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 = (ell.ax ** 2 * np.sin(lamb) ** 2 + ell.ay ** 2 * np.cos(lamb) ** 2) / (
+        LAMBDA__ = (2.0 * b2 * Ee4 * (s2l * s2l)) / (denL * denL * denL) - (2.0 * b2 * Ee2 * s2l) / (
-            ell.Ex ** 2 - ell.Ee ** 2 * np.cos(lamb) ** 2
+            denL * denL
         )
 
-        BETA_ = (ell.ax ** 2 * ell.Ey ** 2 * np.sin(2 * beta)) / (
+        Q = Ey2 * cb2 + Ee2 * sl2
-            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
 
-        BETA__ = (
+        E = BETA * Q
-            (2 * ell.ax ** 2 * ell.Ey ** 4 * np.sin(2 * beta) ** 2)
+        G = LAMBDA * Q
-            / (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)
+        E_beta = BETA_ * Q - BETA * Ey2 * s2b
-        G = LAMBDA * (ell.Ey ** 2 * np.cos(beta) ** 2 + ell.Ee ** 2 * np.sin(lamb) ** 2)
+        E_lamb = BETA * Ee2 * s2l
 
-        E_beta = (
+        G_beta = -LAMBDA * Ey2 * s2b
-            BETA_ * (ell.Ey ** 2 * np.cos(beta) ** 2 + ell.Ee ** 2 * np.sin(lamb) ** 2)
+        G_lamb = LAMBDA_ * Q + LAMBDA * Ee2 * s2l
-            - 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)
+        E_beta_beta = BETA__ * Q - 2.0 * BETA_ * Ey2 * s2b - 2.0 * BETA * Ey2 * c2b
-        G_lamb = (
+        E_beta_lamb = BETA_ * Ee2 * s2l
-            LAMBDA_ * (ell.Ey ** 2 * np.cos(beta) ** 2 + ell.Ee ** 2 * np.sin(lamb) ** 2)
+        E_lamb_lamb = 2.0 * BETA * Ee2 * c2l
-            + LAMBDA * ell.Ee ** 2 * np.sin(2 * lamb)
-        )
 
-        E_beta_beta = (
+        G_beta_beta = -2.0 * LAMBDA * Ey2 * c2b
-            BETA__ * (ell.Ey ** 2 * np.cos(beta) ** 2 + ell.Ee ** 2 * np.sin(lamb) ** 2)
+        G_beta_lamb = -LAMBDA_ * Ey2 * s2b
-            - 2 * BETA_ * ell.Ey ** 2 * np.sin(2 * beta)
+        G_lamb_lamb = LAMBDA__ * Q + 2.0 * LAMBDA_ * Ee2 * s2l + 2.0 * LAMBDA * Ee2 * c2l
-            - 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,
@@ -167,7 +178,7 @@ def gha2_num(
         )
         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)
+        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):
@@ -220,15 +231,12 @@ def gha2_num(
         (_, _, E, G, *_) = BETA_LAMBDA(beta, lamb)
         return np.sqrt(E + G * lamb_p**2)
 
-    # Fall 1 (lambda_0 != lambda_1)
+    def solve_lambda_branch(beta0, lamb0, beta1, lamb1_target, N_run, N_newt, it_max):
-    if abs(lamb_1 - lamb_0) >= 1e-15:
+        dlamb = float(lamb1_target - lamb0)
-        N = int(n)
+        if abs(dlamb) < 1e-15:
-        dlamb = float(lamb_1 - lamb_0)
+            return None
 
-        beta0 = float(beta_0)
+        sgn = 1.0 if dlamb >= 0.0 else -1.0
-        lamb0 = float(lamb_0)
-        beta1 = float(beta_1)
-        lamb1 = float(lamb_1)
 
         def ode_lamb(lamb, v):
             beta, beta_p, X3, X4 = v
@@ -237,16 +245,188 @@ def gha2_num(
             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
+            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)
+        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)
 
-        N_newton = min(N, 4000)
-
         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)
@@ -262,34 +442,33 @@ def gha2_num(
                     return False, None
 
                 step = delta / X3_end
-                step = np.clip(step, -0.5, 0.5)
+                step = float(np.clip(step, -0.5, 0.5))
                 beta_p0 -= step
 
             return False, None
 
-        ok, beta_p0_sol = solve_newton(beta_p0_sph)
+        ok, beta_p0_sol = solve_newton_refine(beta_p0_init)
 
         if not ok:
-            candidates = [-beta_p0_sph, 0.5 * beta_p0_sph, 2.0 * beta_p0_sph]
+            seeds = [beta_p0_sph, -beta_p0_sph, 0.5 * beta_p0_sph, 2.0 * beta_p0_sph]
-            N_quick = min(N, 2000)
             best = None
-            for g in candidates:
+            for seed in seeds:
-                ok_g, sol = solve_newton(g)
+                ok_s, sol_s = solve_newton_refine(seed)
-                if not ok_g:
+                if not ok_s:
                     continue
-                v0_g = np.array([beta0, sol, 0.0, 1.0], dtype=float)
+                v0_s = np.array([beta0, sol_s, 0.0, 1.0], dtype=float)
-                _, _, s_quick = rk4_integral(ode_lamb, lamb0, v0_g, dlamb, N_quick, integrand_lambda)
+                _, _, s_s = rk4_integral(ode_lamb, lamb0, v0_s, dlamb, min(N_full, 2000), integrand_lambda)
-                if (best is None) or (s_quick < best[0]):
+                if (best is None) or (s_s < best[0]):
-                    best = (s_quick, sol)
+                    best = (float(s_s), float(sol_s))
             if best is None:
-                raise RuntimeError("Keine Startwert-Variante konvergiert (lambda-Fall).")
+                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, False)
+            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)
@@ -297,34 +476,35 @@ def gha2_num(
             (_, _, 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_1 = norm_a(arccot(np.sqrt(E_start / G_start) * beta_p_arr[0]))
+            alpha_0 = azimut(E_start, G_start, dbeta_du=beta_p_arr[0] * sgn, dlamb_du=1.0 * sgn)
-            alpha_2 = norm_a(arccot(np.sqrt(E_end   / G_end)   * beta_p_arr[-1]))
+            alpha_1 = azimut(E_end, G_end, dbeta_du=beta_p_arr[-1] * sgn, dlamb_du=1.0 * sgn)
 
-            # Distanz aus Arrays
+            integrand = np.zeros(N_full + 1, dtype=float)
-            integrand = np.zeros(N + 1, dtype=float)
+            for i in range(N_full + 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
+            h = abs(dlamb) / N_full
-            if N % 2 == 0:
+            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 = 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(alpha_1), float(alpha_2), float(s), beta_arr, lamb_arr
+            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, integrand_lambda)
+        _, 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)
-        (_, _, E_end,   G_end,   *_) = BETA_LAMBDA(beta1, lamb1)
+        alpha_0 = azimut(E_start, G_start, dbeta_du=beta_p0 * sgn, dlamb_du=1.0 * sgn)
 
-        alpha_1 = norm_a(arccot(np.sqrt(E_start / G_start) * beta_p0))
+        (_, _, E_end, G_end, *_) = BETA_LAMBDA(float(beta_end), float(lamb0 + dlamb))
-        alpha_2 = norm_a(arccot(np.sqrt(E_end   / G_end)   * beta_p_end))
+        alpha_1 = azimut(E_end, G_end, dbeta_du=float(beta_p_end) * sgn, dlamb_du=1.0 * sgn)
 
-        return float(alpha_1), float(alpha_2), float(s)
+        return float(wrap_0_2pi(alpha_0)), float(wrap_0_2pi(alpha_1)), float(s)
 
     # Fall 2 (lambda_0 == lambda_1)
     N = int(n)
@@ -335,10 +515,7 @@ def gha2_num(
             return 0.0, 0.0, 0.0, np.array([]), np.array([])
         return 0.0, 0.0, 0.0
 
-    beta0 = float(beta_0)
+    sgn = 1.0 if dbeta >= 0.0 else -1.0
-    lamb0 = float(lamb_0)
-    beta1 = float(beta_1)
-    lamb1 = float(lamb_1)
 
     def ode_beta(beta, v):
         lamb, lamb_p, Y3, Y4 = v
@@ -347,10 +524,13 @@ def gha2_num(
         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
+        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 = 0.0
+    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)
@@ -362,10 +542,10 @@ def gha2_num(
             break
 
         if abs(Y3_end) < 1e-20:
-            raise RuntimeError("Abbruch (Ableitung ~ 0) im beta-Fall.")
+            raise RuntimeError("GHA2_num: Ableitung ~ 0 im beta-Fall")
 
         step = delta / Y3_end
-        step = np.clip(step, -1.0, 1.0)
+        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)
@@ -376,11 +556,11 @@ def gha2_num(
         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)
 
-        (BETA_s, LAMBDA_s, _, _, *_) = BETA_LAMBDA(beta_arr[0],  lamb_arr[0])
+        (_, _, E_start, G_start, *_) = BETA_LAMBDA(beta_arr[0], lamb_arr[0])
-        (BETA_e, LAMBDA_e, _, _, *_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
+        (_, _, E_end, G_end, *_) = BETA_LAMBDA(beta_arr[-1], lamb_arr[-1])
 
-        alpha_1 = norm_a((np.pi/2.0) - arccot(np.sqrt(LAMBDA_s / BETA_s) * lamb_p_arr[0]))
+        alpha_0 = azimut(E_start, G_start, dbeta_du=1.0 * sgn, dlamb_du=lamb_p_arr[0] * sgn)
-        alpha_2 = norm_a((np.pi/2.0) - arccot(np.sqrt(LAMBDA_e / BETA_e) * lamb_p_arr[-1]))
+        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):
@@ -389,72 +569,75 @@ def gha2_num(
 
         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 = 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(alpha_1), float(alpha_2), float(s), beta_arr, lamb_arr
+        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
 
-    (BETA_s, LAMBDA_s, _, _, *_) = BETA_LAMBDA(beta0, lamb0)
+    (_, _, E_start, G_start, *_) = BETA_LAMBDA(beta0, lamb0)
-    (BETA_e, LAMBDA_e, _, _, *_) = BETA_LAMBDA(beta1, lamb1)
+    alpha_0 = azimut(E_start, G_start, dbeta_du=1.0 * sgn, dlamb_du=lamb_p0 * sgn)
 
-    alpha_1 = norm_a((np.pi/2.0) - arccot(np.sqrt(LAMBDA_s / BETA_s) * lamb_p0))
+    (_, _, E_end, G_end, *_) = BETA_LAMBDA(beta1, float(lamb_end))
-    alpha_2 = norm_a((np.pi/2.0) - arccot(np.sqrt(LAMBDA_e / BETA_e) * lamb_p_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)
 
-    return float(alpha_1), float(alpha_2), float(s)
 
 if __name__ == "__main__":
-    # ell = EllipsoidTriaxial.init_name("BursaSima1980round")
+    ell = EllipsoidTriaxial.init_name("BursaSima1980round")
-    # beta1 = np.deg2rad(75)
+    beta1 = np.deg2rad(75)
-    # lamb1 = np.deg2rad(-90)
+    lamb1 = np.deg2rad(-90)
-    # beta2 = np.deg2rad(75)
+    beta2 = np.deg2rad(75)
-    # lamb2 = np.deg2rad(66)
+    lamb2 = np.deg2rad(66)
-    # a0, a1, s = gha2_num(ell, beta1, lamb1, beta2, lamb2, n=5000)
+    a0, a1, s = gha2_num(ell, beta1, lamb1, beta2, lamb2, n=100)
-    # print(aus.gms("a0", a0, 4))
+    print(aus.gms("a0", a0, 4))
-    # print(aus.gms("a1", a1, 4))
+    print(aus.gms("a1", a1, 4))
-    # print("s: ", s)
+    print("s: ", s)
-    # # print(aus.gms("a2", a2, 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)
+    cart1 = ell.para2cart(0, 0)
+    cart2 = ell.para2cart(0.4, 1.4)
+    beta1, lamb1 = ell.cart2ell(cart1)
+    beta2, lamb2 = ell.cart2ell(cart2)
 
-    # ell = EllipsoidTriaxial.init_name("BursaSima1980round")
+    a1, a2, s = gha2_num(ell, beta1, lamb1, beta2, lamb2, n=5000)
-    # diffs_panou = []
+    print(s)
-    # examples_panou = ne_panou.get_random_examples(4)
+
-    # for example in examples_panou:
+    ell = EllipsoidTriaxial.init_name("BursaSima1980round")
-    #     beta0, lamb0, beta1, lamb1, _, alpha0, alpha1, s = example
+    diffs_panou = []
-    #     P0 = ell.ell2cart(beta0, lamb0)
+    examples_panou = ne_panou.get_random_examples(4)
-    #     try:
+    for example in examples_panou:
-    #         alpha0_num, alpha1_num, s_num = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=4000, iter_max=10)
+        beta0, lamb0, beta1, lamb1, _, alpha0, alpha1, s = example
-    #         diffs_panou.append(
+        P0 = ell.ell2cart(beta0, lamb0)
-    #             (wu.rad2deg(abs(alpha0 - alpha0_num)), wu.rad2deg(abs(alpha1 - alpha1_num)), abs(s - s_num)))
+        try:
-    #     except:
+            alpha0_num, alpha1_num, s_num = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=4000, iter_max=10)
-    #         print(f"Fehler für {beta0}, {lamb0}, {beta1}, {lamb1}")
+            diffs_panou.append(
-    # diffs_panou = np.array(diffs_panou)
+                (wu.rad2deg(abs(alpha0 - alpha0_num)), wu.rad2deg(abs(alpha1 - alpha1_num)), abs(s - s_num)))
-    # print(diffs_panou)
+        except:
-    #
+            print(f"Fehler für {beta0}, {lamb0}, {beta1}, {lamb1}")
-    # ell = EllipsoidTriaxial.init_name("KarneyTest2024")
+    diffs_panou = np.array(diffs_panou)
-    # diffs_karney = []
+    print(diffs_panou)
-    # # examples_karney = ne_karney.get_examples((30500, 40500))
+
-    # examples_karney = ne_karney.get_random_examples(2)
+    ell = EllipsoidTriaxial.init_name("KarneyTest2024")
-    # for example in examples_karney:
+    diffs_karney = []
-    #     beta0, lamb0, alpha0, beta1, lamb1, alpha1, s = example
+    # examples_karney = ne_karney.get_examples((30500, 40500))
-    #
+    examples_karney = ne_karney.get_random_examples(2)
-    #     try:
+    for example in examples_karney:
-    #         alpha0_num, alpha1_num, s_num = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=4000, iter_max=10)
+        beta0, lamb0, alpha0, beta1, lamb1, alpha1, s = example
-    #         diffs_karney.append((wu.rad2deg(abs(alpha0-alpha0_num)), wu.rad2deg(abs(alpha1-alpha1_num)), abs(s-s_num)))
+
-    #     except:
+        try:
-    #         print(f"Fehler für {beta0}, {lamb0}, {beta1}, {lamb1}")
+            alpha0_num, alpha1_num, s_num = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=4000, iter_max=10)
-    # diffs_karney = np.array(diffs_karney)
+            diffs_karney.append((wu.rad2deg(abs(alpha0-alpha0_num)), wu.rad2deg(abs(alpha1-alpha1_num)), abs(s-s_num)))
-    # print(diffs_karney)
+        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(r"C:\Users\moell\OneDrive\Desktop\Vorlesungen\Master-Projekt\Python_Masterprojekt\GHA_triaxial\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/utils.py

@@ -1,11 +1,13 @@
+from __future__ import annotations
+
 from typing import Tuple
 
 import numpy as np
-from numpy import arctan2, sin, cos, sqrt
+from numpy import arctan2, cos, sin, sqrt
-from numpy._typing import NDArray
 from numpy.typing import NDArray
 
-from ellipsoide import EllipsoidTriaxial
+from ellipsoid_triaxial import EllipsoidTriaxial
+from utils_angle import wrap_0_2pi
 
 
 def sigma2alpha(ell: EllipsoidTriaxial, sigma: NDArray, point: NDArray) -> float:
@@ -21,7 +23,7 @@ def sigma2alpha(ell: EllipsoidTriaxial, sigma: NDArray, point: NDArray) -> float
     Q = float(q @ sigma)
 
     alpha = arctan2(P, Q)
-    return alpha
+    return wrap_0_2pi(alpha)
 
 
 def alpha_para2ell(ell: EllipsoidTriaxial, u: float, v: float, alpha_para: float) -> Tuple[float, float, float]:
@@ -43,10 +45,10 @@ def alpha_para2ell(ell: EllipsoidTriaxial, u: float, v: float, alpha_para: float
     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:
+    if np.linalg.norm(sigma_para - sigma_ell) > 1e-7:
-        raise Exception("Alpha Umrechnung fehlgeschlagen")
+        raise Exception("alpha_para2ell: Differenz in den Richtungsableitungen")
 
-    return beta, lamb, alpha_ell
+    return beta, lamb, wrap_0_2pi(alpha_ell)
 
 
 def alpha_ell2para(ell: EllipsoidTriaxial, beta: float, lamb: float, alpha_ell: float) -> Tuple[float, float, float]:
@@ -68,10 +70,10 @@ def alpha_ell2para(ell: EllipsoidTriaxial, beta: float, lamb: float, alpha_ell:
     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-9:
+    if np.linalg.norm(sigma_para - sigma_ell) > 1e-7:
-        raise Exception("Alpha Umrechnung fehlgeschlagen")
+        raise Exception("alpha_ell2para: Differenz in den Richtungsableitungen")
 
-    return u, v, alpha_para
+    return u, v, wrap_0_2pi(alpha_para)
 
 
 def func_sigma_ell(ell: EllipsoidTriaxial, point: NDArray, alpha_ell: float) -> NDArray:
@@ -124,18 +126,19 @@ def pq_ell(ell: EllipsoidTriaxial, point: NDArray) -> Tuple[NDArray, NDArray]:
     :param point: Punkt
     :return: p und q
     """
-    x, y, z = point
     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
 
-    c1 = x ** 2 + y ** 2 + z ** 2 - (ell.ax ** 2 + ell.ay ** 2 + ell.b ** 2)
+        _, t2 = ell.func_t12(point)
-    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)
@@ -171,13 +174,28 @@ def pq_para(ell: EllipsoidTriaxial, point: NDArray) -> Tuple[NDArray, NDArray]:
                   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")
-
     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

Tests/test_biaxial.py

@@ -1,34 +0,0 @@
-import numpy as np
-from ellipsoide import EllipsoidBiaxial
-from GHA_biaxial.bessel import gha1 as gha1_bessel
-from GHA_biaxial.gauss import gha1 as gha1_gauss
-from GHA_biaxial.rk import gha1 as gha1_rk
-from GHA_biaxial.gauss import gha2 as gha2_gauss
-
-re = EllipsoidBiaxial.init_name("Bessel")
-
-# phi0 = 0.6
-# lamb0 = 1.2
-# alpha0 = 0.45
-# s = 123456
-#
-# values_bessel = gha1_bessel(re, phi0, lamb0, alpha0, s)
-# alpha1_bessel = values_bessel[-1]
-# p1_bessel = re.bi_ell2cart(values_bessel[0], values_bessel[1], 0)
-#
-# values_gauss1 = gha1_gauss(re, phi0, lamb0, alpha0, s)
-# alpha1_gauss1 = values_gauss1[-1]
-# p1_gauss = re.bi_ell2cart(values_gauss1[0], values_gauss1[1], 0)
-#
-# values_rk = gha1_rk(re, phi0, lamb0 , alpha0, s, 10000)
-# alpha1_rk = values_rk[-1]
-# p1_rk = re.bi_ell2cart(values_rk[0], values_rk[1], 0)
-#
-# alpha0_gauss, alpha1_gauss2, s_gauss = gha2_gauss(re, phi0, lamb0, values_gauss1[0], values_gauss1[1])
-
-phi0 = 0.6
-lamb0 = 1.2
-
-cart = re.bi_ell2cart(phi0, lamb0, 0)
-ell = re.bi_cart2ell(cart)
-pass

ausgaben.py

@@ -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)
+    values = wu.rad2gms(rad)
-    return f"{name} = {int(gms[0])}° {int(gms[1])}' {round(gms[2],stellen):.{stellen}f}''"
+    return f"{name} = {int(values[0])}° {int(values[1])}' {round(values[2], stellen):.{stellen}f}''"

ellipsoid_triaxial.py

@@ -1,116 +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
-from typing import Tuple
-from numpy.typing import NDArray
 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
 
-class EllipsoidBiaxial:
+import jacobian_Ligas
-    def __init__(self, a: float, b: float):
+from utils_angle import wrap_mhalfpi_halfpi, wrap_mpi_pi
-        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: np.arctan2(self.a * np.sin(beta), self.b * np.cos(beta))
-    beta2phi = lambda self, beta: np.arctan2(self.a ** 2 * np.sin(beta), self.b ** 2 * np.cos(beta))
-
-    psi2beta = lambda self, psi: np.arctan2(self.b * np.sin(psi), self.a * np.cos(psi))
-    psi2phi = lambda self, psi: np.arctan2(self.a * np.sin(psi), self.b * np.cos(psi))
-
-    phi2beta = lambda self, phi: np.arctan2(self.b**2 * np.sin(phi), self.a**2 * np.cos(phi))
-    phi2psi = lambda self, phi: np.arctan2(self.b * np.sin(phi), self.a * np.cos(phi))
-
-    phi2p = lambda self, phi: self.N(phi) * cos(phi)
-
-    def bi_cart2ell(self, point: NDArrayself, 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 = 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])
 
 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
@@ -124,14 +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, BursaSima1980round, Eitschberger1978, Bursa1972,
+        Mögliche Ellipsoide: BursaSima1980round, KarneyTest2024, Fiction, BursaFialova1993, BursaSima1980, Eitschberger1978, Bursa1972,
-        Bursa1970, BesselBiaxial, Fiction, KarneyTest2024
+        Bursa1970
         Panou et al (2020)
         :param name: Name des dreiachsigen Ellipsoids
+        :return: dreiachsiger Ellipsoid
         """
         if name == "BursaFialova1993":
             ax = 6378171.36
@@ -164,11 +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
@@ -179,6 +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:
         """
@@ -218,8 +124,10 @@ 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)
-        if c1 ** 2 - 4 * c0 < 0:
+        if c1 ** 2 - 4 * c0 < -1e-9:
-            t2 = np.nan
+            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:
@@ -261,7 +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:
@@ -284,6 +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
 
@@ -412,14 +324,14 @@ class EllipsoidTriaxial:
                 i += 1
 
             if i == maxI:
-                raise Exception("Umrechnung ist nicht konvergiert")
+                raise Exception("Umrechnung cart2ell: nicht konvergiert")
 
             point_n = self.ell2cart(beta, lamb)
             delta_r = np.linalg.norm(point - point_n, axis=-1)
             if delta_r > 1e-6:
-                raise Exception("Fehler in der Umrechnung cart2ell")
+                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
@@ -511,7 +423,7 @@ class EllipsoidTriaxial:
             i += 1
 
         if i == maxI:
-            raise Exception("Umrechung ist nicht konvergiert")
+            raise Exception("Umrechnung cart2ell: nicht konvergiert")
 
         return phi, lamb
 
@@ -577,6 +489,8 @@ 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)
@@ -598,15 +512,15 @@ class EllipsoidTriaxial:
                 phi, lamb, h = self.cart2geod(point, f"ligas{new_mode}", maxIter, maxLoa)
             else:
                 if xG < 0 and yG < 0:
-                    lamb = -pi + lamb
+                    lamb += -pi
 
                 elif xG < 0:
-                    lamb = pi + lamb
+                    lamb += pi
 
                 if abs(zG) < eps:
                     phi = 0
-
+        wrap_mhalfpi_halfpi(phi), wrap_mpi_pi(lamb)
-        return phi, lamb, h
+        return wrap_mhalfpi_halfpi(phi), wrap_mpi_pi(lamb), h
 
     def para2cart(self, u: float | NDArray, v: float | NDArray) -> NDArray:
         """
@@ -643,8 +557,8 @@ class EllipsoidTriaxial:
             v = 2 * arctan2(v_check1, v_check2 + v_factor)
         else:
             v = pi/2 - 2 * arctan2(v_check2, v_check1 + v_factor)
-
+        wrap_mhalfpi_halfpi(u), wrap_mpi_pi(v)
-        return u, v
+        return wrap_mhalfpi_halfpi(u), wrap_mpi_pi(v)
 
     def ell2para(self, beta: float, lamb: float) -> Tuple[float, float]:
         """
@@ -749,63 +663,71 @@ class EllipsoidTriaxial:
 
 
 if __name__ == "__main__":
-    ell = EllipsoidTriaxial.init_name("BursaSima1980")
+    ell = EllipsoidTriaxial.init_name("KarneyTest2024")
-    diff_list = []
+    # cart = ell.ell2cart(pi/2, 0)
-    diffs_para = []
+    # print(cart)
-    diffs_ell = []
+    # cart = ell.ell2cart(pi/2*8999999999999999/9000000000000000, 0)
-    diffs_geod = []
+    # print(cart)
-    points = []
+    elli = ell.cart2ell(np.array([0, 0.0, 1/sqrt(2)]))
-    for v_deg in range(-180, 181, 5):
+    print(elli)
-        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)
+    # ell = EllipsoidTriaxial.init_name("BursaSima1980")
-            cart_elli = ell.ell2cart(elli[0], elli[1])
+    # diff_list = []
-            diff_ell = np.linalg.norm(point - cart_elli, axis=-1)
+    # diffs_para = []
-
+    # diffs_ell = []
-            para = ell.cart2para(point)
+    # diffs_geod = []
-            cart_para = ell.para2cart(para[0], para[1])
+    # points = []
-            diff_para = np.linalg.norm(point - cart_para, axis=-1)
+    # for v_deg in range(-180, 181, 5):
-
+    #     for u_deg in range(-90, 91, 5):
-            geod = ell.cart2geod(point, "ligas3")
+    #         v = wu.deg2rad(v_deg)
-            cart_geod = ell.geod2cart(geod[0], geod[1], geod[2])
+    #         u = wu.deg2rad(u_deg)
-            diff_geod3 = np.linalg.norm(point - cart_geod, axis=-1)
+    #         point = ell.para2cart(u, v)
-
+    #         points.append(point)
-            diff_list.append([v_deg, u_deg, diff_ell, diff_para, diff_geod3])
+    #
-            diffs_ell.append([diff_ell])
+    #         elli = ell.cart2ell(point)
-            diffs_para.append([diff_para])
+    #         cart_elli = ell.ell2cart(elli[0], elli[1])
-            diffs_geod.append([diff_geod3])
+    #         diff_ell = np.linalg.norm(point - cart_elli, axis=-1)
-
+    #
-    diff_list = np.array(diff_list)
+    #         para = ell.cart2para(point)
-    diffs_ell = np.array(diffs_ell)
+    #         cart_para = ell.para2cart(para[0], para[1])
-    diffs_para = np.array(diffs_para)
+    #         diff_para = np.linalg.norm(point - cart_para, axis=-1)
-    diffs_geod = np.array(diffs_geod)
+    #
-
+    #         geod = ell.cart2geod(point, "ligas3")
-    pass
+    #         cart_geod = ell.geod2cart(geod[0], geod[1], geod[2])
-
+    #         diff_geod3 = np.linalg.norm(point - cart_geod, axis=-1)
-    points = np.array(points)
+    #
-    fig = plt.figure()
+    #         diff_list.append([v_deg, u_deg, diff_ell, diff_para, diff_geod3])
-    ax = fig.add_subplot(projection='3d')
+    #         diffs_ell.append([diff_ell])
-
+    #         diffs_para.append([diff_para])
-    sc = ax.scatter(
+    #         diffs_geod.append([diff_geod3])
-        points[:, 0],
+    #
-        points[:, 1],
+    # diff_list = np.array(diff_list)
-        points[:, 2],
+    # diffs_ell = np.array(diffs_ell)
-        c=diffs_ell,  # Farbcode = diff
+    # diffs_para = np.array(diffs_para)
-        cmap='viridis',  # Colormap
+    # diffs_geod = np.array(diffs_geod)
-        s=10 + 20 * diffs_ell,  # optional: Größe abhängig vom diff
+    #
-        alpha=0.8
+    # pass
-    )
+    #
-
+    # points = np.array(points)
-    # Farbskala
+    # fig = plt.figure()
-    cbar = plt.colorbar(sc)
+    # ax = fig.add_subplot(projection='3d')
-    cbar.set_label("diff")
+    #
-
+    # sc = ax.scatter(
-    ax.set_xlabel("X")
+    #     points[:, 0],
-    ax.set_ylabel("Y")
+    #     points[:, 1],
-    ax.set_zlabel("Z")
+    #     points[:, 2],
-
+    #     c=diffs_ell,  # Farbcode = diff
-    plt.show()
+    #     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 +1,8 @@
+from typing import Tuple
+
 import numpy as np
 from numpy.typing import NDArray
-from typing import Tuple
+
 
 def case1(E: float, F: float, G: float, pG: NDArray, pE: NDArray) -> Tuple[NDArray, NDArray]:
     """
@@ -34,7 +36,7 @@ def case1(E: float, F: float, G: float, pG: NDArray, pE: NDArray) -> Tuple[NDArr
 
     return invJ, fxE
 
-def case2(E: float, F: float, G: float, pG: np.ndarray, pE: np.ndarray) -> Tuple[NDArray, 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
@@ -68,7 +70,7 @@ def case2(E: float, F: float, G: float, pG: np.ndarray, pE: np.ndarray) -> Tuple
 
     return invJ, fxE
 
-def case3(E: float, F: float, G: float, pG: np.ndarray, pE: np.ndarray) -> Tuple[NDArray, 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

nicht abgeben/GHA_biaxial/bessel.py

@@ -1,10 +1,20 @@
-from numpy import *
-import scipy as sp
-from ellipsoide import EllipsoidBiaxial
 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))

nicht abgeben/GHA_biaxial/gauss.py

@@ -1,8 +1,8 @@
-from numpy import sin, cos, pi, sqrt, tan, arcsin, arccos, arctan
-import ausgaben as aus
-from ellipsoide import EllipsoidBiaxial
 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]:
     """

nicht abgeben/GHA_biaxial/rk.py

@@ -1,13 +1,26 @@
-import runge_kutta as rk
+from typing import Tuple
-from numpy import sin, cos, tan
+
-import winkelumrechnungen as wu
-from ellipsoide import EllipsoidBiaxial
 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, v):
+        def ODE(s: float, v: NDArray):
             phi, lam, A = v
             V = re.V(phi)
             dphi = cos(A) * V ** 3 / re.c

nicht abgeben/Tests/alpha_conversion_test.ipynb

@@ -1,56 +1,41 @@
 {
  "cells": [
   {
+   "metadata": {},
    "cell_type": "code",
-   "id": "initial_id",
-   "metadata": {
-    "collapsed": true,
-    "ExecuteTime": {
-     "end_time": "2026-01-20T15:30:31.978159Z",
-     "start_time": "2026-01-20T15:30:31.835157Z"
-    }
-   },
    "source": [
     "%load_ext autoreload\n",
     "%autoreload 2"
    ],
+   "id": "a78faf7f4883772f",
    "outputs": [],
-   "execution_count": 1
+   "execution_count": null
   },
   {
-   "metadata": {
+   "metadata": {},
-    "ExecuteTime": {
-     "end_time": "2026-01-20T15:30:33.910807Z",
-     "start_time": "2026-01-20T15:30:32.803089Z"
-    }
-   },
    "cell_type": "code",
    "source": [
     "%reload_ext autoreload\n",
     "%autoreload 2\n",
+    "import numpy as np\n",
+    "\n",
     "import winkelumrechnungen as wu\n",
-    "from ellipsoide import EllipsoidTriaxial\n",
+    "from GHA_triaxial.utils import alpha_ell2para, alpha_para2ell\n",
-    "from GHA_triaxial.utils import alpha_para2ell, alpha_ell2para\n",
+    "from ellipsoid_triaxial import EllipsoidTriaxial"
-    "import numpy as np"
    ],
-   "id": "9ad815aea55574e3",
+   "id": "46aa84a937fea491",
    "outputs": [],
-   "execution_count": 2
+   "execution_count": null
   },
   {
-   "metadata": {
+   "metadata": {},
-    "ExecuteTime": {
-     "end_time": "2026-01-20T15:33:40.785362Z",
-     "start_time": "2026-01-20T15:33:34.296487Z"
-    }
-   },
    "cell_type": "code",
    "source": [
     "ell = EllipsoidTriaxial.init_name(\"KarneyTest2024\")\n",
     "diffs = []\n",
-    "for beta_deg in range(-180, 181, 45):\n",
+    "for beta_deg in range(-90, 91, 15):\n",
-    "    for lamb_deg in range(-90, 91, 45):\n",
+    "    for lamb_deg in range(-180, 180, 15):\n",
-    "        for alpha_deg in range(0, 360, 45):\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",
@@ -67,17 +52,12 @@
     "            diffs.append((beta_deg, lamb_deg, alpha_deg, diff_1, diff_2))\n",
     "diffs = np.array(diffs)"
    ],
-   "id": "98b9b220118deb3f",
+   "id": "82fc6cbbe7d5abcb",
    "outputs": [],
-   "execution_count": 6
+   "execution_count": null
   },
   {
-   "metadata": {
+   "metadata": {},
-    "ExecuteTime": {
-     "end_time": "2026-01-20T15:33:50.497990Z",
-     "start_time": "2026-01-20T15:33:50.261115Z"
-    }
-   },
    "cell_type": "code",
    "source": [
     "i_max_ell = np.argmax(diffs[:, 3])\n",
@@ -92,18 +72,9 @@
     "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": "3c74b65b0e85e3c2",
+   "id": "97b5b8c9ca5377ab",
-   "outputs": [
+   "outputs": [],
-    {
+   "execution_count": null
-     "name": "stdout",
-     "output_type": "stream",
-     "text": [
-      "Für elliptisches Alpha = 315.0° und beta = -90.0°, lamb = -90.0°: diff = 3.426945967752335e-05\"\n",
-      "Für parametrisches Alpha = 315.0° und beta = -90.0°, lamb = -90.0°: diff = 3.426945967752335e-05\"\n"
-     ]
-    }
-   ],
-   "execution_count": 7
   }
  ],
  "metadata": {

nicht abgeben/Tests/conversions_test.ipynb

@@ -20,13 +20,14 @@
    "source": [
     "%reload_ext autoreload\n",
     "%autoreload 2\n",
-    "import pickle\n",
-    "import numpy as np\n",
-    "import winkelumrechnungen as wu\n",
     "from itertools import product\n",
+    "\n",
+    "import numpy as np\n",
     "import pandas as pd\n",
-    "from ellipsoide import EllipsoidTriaxial\n",
+    "import plotly.graph_objects as go\n",
-    "import plotly.graph_objects as go"
+    "\n",
+    "import winkelumrechnungen as wu\n",
+    "from ellipsoid_triaxial import EllipsoidTriaxial"
    ],
    "outputs": [],
    "execution_count": null

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

@@ -1,7 +1,9 @@
-import numpy as np
-from numpy import sqrt, arctan2, sin, cos, arcsin, arccos
-from numpy.typing import NDArray
 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
 
 
@@ -77,7 +79,7 @@ def gha2(R: float, phi0: float, lamb0: float, phi1: float, lamb1: float) -> Tupl
     alpha1 = arctan2(-cos(phi0) * sin(lamb1 - lamb0),
                      cos(phi1) * sin(phi0) - sin(phi1) * cos(phi0) * cos(lamb1 - lamb0))
     if alpha1 < 0:
-        alpha1 += 2 * np.pi
+        alpha1 += 2 * pi
 
     return alpha0, alpha1, s
 

nicht abgeben/plots.ipynb

@@ -9,10 +9,11 @@
    },
    "cell_type": "code",
    "source": [
-    "import plotly.graph_objects as go\n",
     "import numpy as np\n",
-    "from ellipsoide import EllipsoidTriaxial\n",
+    "import plotly.graph_objects as go\n",
-    "import winkelumrechnungen as wu"
+    "\n",
+    "import winkelumrechnungen as wu\n",
+    "from ellipsoid_triaxial import EllipsoidTriaxial"
    ],
    "id": "731173e4745cfe7c",
    "outputs": [],

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
@@ -35,14 +38,32 @@ 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, t0: float, v0: np.ndarray, weite: float, schritte: int, fein: bool = False):
+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)
@@ -62,8 +83,19 @@ def rk4_end(ode, t0: float, v0: np.ndarray, weite: float, schritte: int, fein: b
     return t, v
 
 # RK4 mit Simpson bzw. Trapez
-def rk4_integral( ode, t0: float, v0: np.ndarray, weite: float, schritte: int, integrand_at, fein: bool = False, simpson: bool = True, ):
+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)
 

test.py

@@ -1,7 +0,0 @@
-import numpy as np
-import ellipsoide
-
-ell = ellipsoide.EllipsoidTriaxial.init_name("KarneyTest2024")
-
-cart = ell.para2cart(0, np.pi/2)
-print(cart)

utils_angle.py

@@ -1,13 +1,54 @@
 import numpy as np
 
+import winkelumrechnungen as wu
 
-def arccot(x):
+
+def arccot(x: float) -> float:
+    """
+    Berechnung von arccot eines Winkels
+    :param x: Winkel
+    :return: arccot(Winkel)
+    """
     return np.arctan2(1.0, x)
 
 
-def cot(a):
+def cot(x: float) -> float:
-    return np.cos(a) / np.sin(a)
+    """
+    Berechnung von cot eines Winkels
+    :param x: Winkel
+    :return: cot(Winkel)
+    """
+    return np.cos(x) / np.sin(x)
 
 
-def wrap_to_pi(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))))