Merge remote-tracking branch 'origin/main'
# Conflicts: # GHA_triaxial/gha2_num.py
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Tammo.Weber
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GHA_triaxial/gha1_ES.py
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GHA_triaxial/gha1_ES.py
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from __future__ import annotations
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from typing import List, Optional, Tuple
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import numpy as np
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from ellipsoide import EllipsoidTriaxial
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from GHA_triaxial.gha1_ana import gha1_ana
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from GHA_triaxial.gha1_approx import gha1_approx
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from Hansen_ES_CMA import escma
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from utils_angle import wrap_to_pi
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from numpy.typing import NDArray
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def ellipsoid_formparameter(ell: EllipsoidTriaxial):
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"""
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Berechnet die Formparameter des dreiachsigen Ellipsoiden nach Karney (2025), Gl. (2)
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:param ell: Ellipsoid
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:return: e, k und k'
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"""
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nenner = np.sqrt(max(ell.ax * ell.ax - ell.b * ell.b, 0.0))
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k = np.sqrt(max(ell.ay * ell.ay - ell.b * ell.b, 0.0)) / nenner
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k_ = np.sqrt(max(ell.ax * ell.ax - ell.ay * ell.ay, 0.0)) / nenner
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e = np.sqrt(max(ell.ax * ell.ax - ell.b * ell.b, 0.0)) / ell.ay
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return e, k, k_
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def ENU_beta_omega(beta: float, omega: float, ell: EllipsoidTriaxial) \
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-> Tuple[NDArray, NDArray, NDArray, float, float, NDArray]:
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"""
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Analytische ENU-Basis in ellipsoidische Koordinaten (β, ω) nach Karney (2025), S. 2
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:param beta: Beta Koordinate
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:param omega: Omega Koordinate
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:param ell: Ellipsoid
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:return: E_hat = Einheitsrichtung entlang wachsendem ω (East)
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N_hat = Einheitsrichtung entlang wachsendem β (North)
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U_hat = Einheitsnormale (Up)
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En & Nn = Längen der unnormierten Ableitungen
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R (XYZ) = Punkt in XYZ
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"""
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# Berechnungshilfen
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omega = wrap_to_pi(omega)
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cb = np.cos(beta)
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sb = np.sin(beta)
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co = np.cos(omega)
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so = np.sin(omega)
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# D = sqrt(a^2 - c^2)
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D = np.sqrt(ell.ax*ell.ax - ell.b*ell.b)
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# Sx = sqrt(a^2 - b^2 sin^2β - c^2 cos^2β)
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Sx = np.sqrt(ell.ax*ell.ax - ell.ay*ell.ay*(sb*sb) - ell.b*ell.b*(cb*cb))
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# Sz = sqrt(a^2 sin^2ω + b^2 cos^2ω - c^2)
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Sz = np.sqrt(ell.ax*ell.ax*(so*so) + ell.ay*ell.ay*(co*co) - ell.b*ell.b)
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# Karney Gl. (4)
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X = ell.ax * co * Sx / D
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Y = ell.ay * cb * so
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Z = ell.b * sb * Sz / D
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R = np.array([X, Y, Z], dtype=float)
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# --- Ableitungen - Karney Gl. (5a,b,c)---
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# E = ∂R/∂ω
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dX_dw = -ell.ax * so * Sx / D
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dY_dw = ell.ay * cb * co
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dZ_dw = ell.b * sb * (so * co * (ell.ax*ell.ax - ell.ay*ell.ay) / Sz) / D
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E = np.array([dX_dw, dY_dw, dZ_dw], dtype=float)
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# N = ∂R/∂β
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dX_db = ell.ax * co * (sb * cb * (ell.b*ell.b - ell.ay*ell.ay) / Sx) / D
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dY_db = -ell.ay * sb * so
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dZ_db = ell.b * cb * Sz / D
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N = np.array([dX_db, dY_db, dZ_db], dtype=float)
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# U = Grad(x^2/a^2 + y^2/b^2 + z^2/c^2 - 1)
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U = np.array([X/(ell.ax*ell.ax), Y/(ell.ay*ell.ay), Z/(ell.b*ell.b)], dtype=float)
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En = np.linalg.norm(E)
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Nn = np.linalg.norm(N)
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Un = np.linalg.norm(U)
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N_hat = N / Nn
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E_hat = E / En
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U_hat = U / Un
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return E_hat, N_hat, U_hat, En, Nn, R
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def jacobi_konstante(beta: float, omega: float, alpha: float, ell: EllipsoidTriaxial) -> float:
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"""
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Jacobi-Konstante nach Karney (2025), Gl. (14)
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:param beta: Beta Koordinate
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:param omega: Omega Koordinate
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:param alpha: Azimut alpha
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:param ell: Ellipsoid
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:return: Jacobi-Konstante
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"""
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e, k, k_ = ellipsoid_formparameter(ell)
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gamma_jacobi = float((k ** 2) * (np.cos(beta) ** 2) * (np.sin(alpha) ** 2) - (k_ ** 2) * (np.sin(omega) ** 2) * (np.cos(alpha) ** 2))
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return gamma_jacobi
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def azimuth_at_ESpoint(P_prev: NDArray, P_curr: NDArray, E_hat_curr: NDArray, N_hat_curr: NDArray, U_hat_curr: NDArray) -> float:
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"""
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Berechnet das Azimut in der lokalen Tangentialebene am aktuellen Punkt P_curr, gemessen
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an der Bewegungsrichtung vom vorherigen Punkt P_prev nach P_curr.
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:param P_prev: vorheriger Punkt
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:param P_curr: aktueller Punkt
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:param E_hat_curr: Einheitsvektor der lokalen Tangentialrichtung am Punkt P_curr
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:param N_hat_curr: Einheitsvektor der lokalen Tangentialrichtung am Punkt P_curr
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:param U_hat_curr: Einheitsnormalenvektor am Punkt P_curr
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:return: Azimut in Radiant
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"""
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v = (P_curr - P_prev).astype(float)
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vT = v - float(np.dot(v, U_hat_curr)) * U_hat_curr
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vT_hat = vT / np.linalg.norm(vT)
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sE = float(np.dot(vT_hat, E_hat_curr))
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sN = float(np.dot(vT_hat, N_hat_curr))
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return wrap_to_pi(float(np.arctan2(sE, sN)))
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def optimize_next_point(beta_i: float, omega_i: float, alpha_i: float, ds: float, gamma0: float,
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ell: EllipsoidTriaxial, maxSegLen: float = 1000.0, sigma0: float = None) -> Tuple[float, float, NDArray, float]:
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"""
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Berechnung der 1. GHA mithilfe der CMA-ES.
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Die CMA-ES optimiert sukzessive einen Punkt, der maxSegLen vom vorherigen Punkt entfernt und zusätzlich auf der
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geodätischen Linien liegt. Somit entsteht ein Geodäten ähnlicher Polygonzug auf der Oberfläche des dreiachsigen Ellipsoids.
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:param beta_i: Beta Koordinate am Punkt i
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:param omega_i: Omega Koordinate am Punkt i
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:param alpha_i: Azimut am Punkt i
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:param ds: Gesamtlänge
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:param gamma0: Jacobi-Konstante am Startpunkt
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:param ell: Ellipsoid
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:param maxSegLen: maximale Segmentlänge
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:param sigma0:
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:return:
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"""
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# Startbasis
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E_i, N_i, U_i, En_i, Nn_i, P_i = ENU_beta_omega(beta_i, omega_i, ell)
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# Prediktor: dβ ≈ ds cosα / |N|, dω ≈ ds sinα / |E|
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d_beta = ds * float(np.cos(alpha_i)) / Nn_i
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d_omega = ds * float(np.sin(alpha_i)) / En_i
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beta_pred = beta_i + d_beta
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omega_pred = wrap_to_pi(omega_i + d_omega)
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xmean = np.array([beta_pred, omega_pred], dtype=float)
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if sigma0 is None:
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R0 = (ell.ax + ell.ay + ell.b) / 3
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sigma0 = 1e-5 * (ds / R0)
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def fitness(x: NDArray) -> float:
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"""
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Fitnessfunktion: Fitnesscheck erfolgt anhand der Segmentlänge und der Jacobi-Konstante.
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Die Segmentlänge muss möglichst gut zum Sollwert passen. Die Jacobi-Konstante am Punkt x muss zur
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Jacobi-Konstanten am Startpunkt passen, damit der Polygonzug auf derselben geodätischen Linie bleibt.
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:param x: Koordinate in beta, lambda aus der CMA-ES
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:return: Fitnesswert (f)
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"""
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beta = x[0]
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omega = wrap_to_pi(x[1])
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P = ell.ell2cart(beta, omega) # in kartesischer Koordinaten
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d = float(np.linalg.norm(P - P_i)) # Distanz zwischen
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# maxSegLen einhalten
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J_len = ((d - ds) / ds) ** 2
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w_len = 1.0
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# Azimut für Jacobi-Konstante
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E_j, N_j, U_j, _, _, _ = ENU_beta_omega(beta, omega, ell)
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alpha_end = azimuth_at_ESpoint(P_i, P, E_j, N_j, U_j)
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# Jacobi-Konstante
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g_end = jacobi_konstante(beta, omega, alpha_end, ell)
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J_gamma = (g_end - gamma0) ** 2
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w_gamma = 10
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f = float(w_len * J_len + w_gamma * J_gamma)
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return f
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xb = escma(fitness, N=2, xmean=xmean, sigma=sigma0) # Aufruf CMA-ES
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beta_best = xb[0]
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omega_best = wrap_to_pi(xb[1])
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P_best = ell.ell2cart(beta_best, omega_best)
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E_j, N_j, U_j, _, _, _ = ENU_beta_omega(beta_best, omega_best, ell)
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alpha_end = azimuth_at_ESpoint(P_i, P_best, E_j, N_j, U_j)
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return beta_best, omega_best, P_best, alpha_end
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def gha1_ES(ell: EllipsoidTriaxial, beta0: float, omega0: float, alpha0: float, s_total: float, maxSegLen: float = 1000):
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"""
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Aufruf der 1. GHA mittels CMA-ES
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:param ell: Ellipsoid
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:param beta0: Beta Startkoordinate
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:param omega0: Omega Startkoordinate
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:param alpha0: Azimut Startkoordinate
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:param s_total: Gesamtstrecke
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:param maxSegLen: maximale Segmentlänge
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:return: Zielpunkt Pk und Azimut am Zielpunkt
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"""
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beta = float(beta0)
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omega = wrap_to_pi(float(omega0))
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alpha = wrap_to_pi(float(alpha0))
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gamma0 = jacobi_konstante(beta, omega, alpha, ell) # Referenz-γ0
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points: List[NDArray] = [ell.ell2cart(beta, omega)]
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alpha_end: List[float] = [alpha]
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s_acc = 0.0
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step = 0
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nsteps_est = int(np.ceil(s_total / maxSegLen))
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while s_acc < s_total - 1e-9:
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step += 1
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ds = min(maxSegLen, s_total - s_acc)
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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}")
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beta, omega, P, alpha = optimize_next_point(beta_i=beta, omega_i=omega, alpha_i=alpha, ds=ds, gamma0=gamma0,
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ell=ell, maxSegLen=maxSegLen)
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s_acc += ds
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points.append(P)
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alpha_end.append(alpha)
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if step > nsteps_est + 50:
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raise RuntimeError("Zu viele Schritte – vermutlich Konvergenzproblem / falsche Azimut-Konvention.")
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Pk = points[-1]
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alpha1 = alpha_end[-1]
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return Pk, alpha1
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if __name__ == "__main__":
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ell = EllipsoidTriaxial.init_name("BursaSima1980round")
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s = 188891.650873
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alpha0 = 70/(180/np.pi)
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P0 = ell.ell2cart(5/(180/np.pi), -90/(180/np.pi))
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point1, alpha1 = gha1_ana(ell, P0, alpha0=alpha0, s=s, maxM=100, maxPartCircum=32)
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point1app, alpha1app = gha1_approx(ell, P0, alpha0=alpha0, s=s, ds=1000)
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res, alpha = gha1_ES(ell, beta0=5/(180/np.pi), omega0=-90/(180/np.pi), alpha0=alpha0, s_total=s, maxSegLen=1000)
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print(point1)
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print(res)
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print(alpha)
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# print("alpha1 (am Endpunkt):", res.alpha1)
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print(res - point1)
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print(point1app - point1, "approx")
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@@ -49,7 +49,7 @@ def midpoint_fitness(x: tuple) -> float:
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return f
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def gha2_ES(ell: EllipsoidTriaxial, P0: NDArray, Pk: NDArray, maxSegLen: float = None, stopeval: int = 2000, maxIter: int = 10000, all_points: bool = False):
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def gha2_ES(ell: EllipsoidTriaxial, P0: NDArray, Pk: NDArray, maxSegLen: float = None, maxIter: int = 10000, all_points: bool = False):
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"""
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Berechnen der 2. GHA mithilfe der CMA-ES.
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Die CMA-ES optimiert sukzessive den Mittelpunkt zwischen Start- und Zielpunkt. Der Abbruch der Berechnung erfolgt, wenn alle Segmentlängen <= maxSegLen sind.
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@@ -58,7 +58,6 @@ def gha2_ES(ell: EllipsoidTriaxial, P0: NDArray, Pk: NDArray, maxSegLen: float =
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:param P0: Startpunkt
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:param Pk: Zielpunkt
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:param maxSegLen: maximale Segmentlänge
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:param stopeval: maximale Durchläufe der CMA-ES
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:param maxIter: maximale Durchläufe der Mittelpunktsgenerierung
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:param all_points: Ergebnisliste mit allen Punkte, die wahlweise mit ausgegeben werden kann
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:return: Richtungswinkel des Start- und Zielpunktes und Gesamtlänge
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@@ -67,7 +66,7 @@ def gha2_ES(ell: EllipsoidTriaxial, P0: NDArray, Pk: NDArray, maxSegLen: float =
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ell_ES = ell
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R0 = (ell.ax + ell.ay + ell.b) / 3
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if maxSegLen is None:
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maxSegLen = R0 * 1 / (637.4) # 10km Segment bei mittleren Erdradius
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maxSegLen = R0 * 1 / (637.4*2) # 10km Segment bei mittleren Erdradius
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sigma_uv_nom = 1e-3 * (maxSegLen / R0) # ~1e-5
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@@ -101,8 +100,7 @@ def gha2_ES(ell: EllipsoidTriaxial, P0: NDArray, Pk: NDArray, maxSegLen: float =
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sigmaStep = sigma_uv_nom * (Sehne(A, B) / maxSegLen)
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u, v = escma(midpoint_fitness, N=2, xmean=xmean, sigma=sigmaStep, stopfitness=-np.inf,
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stopeval=stopeval)
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u, v = escma(midpoint_fitness, N=2, xmean=xmean, sigma=sigmaStep)
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P_next = ell.para2cart(u, v)
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new_points.append(P_next)
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@@ -171,7 +169,7 @@ if __name__ == '__main__':
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beta1, lamb1 = (0.7, 0.3)
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P1 = ell.ell2cart(beta1, lamb1)
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alpha0, alpha1, s_num, betas, lambs = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=10000, all_points=True)
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alpha0, alpha1, s_num, betas, lambs = gha2_num(ell, beta0, lamb0, beta1, lamb1, n=1000, all_points=True)
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points_num = []
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for beta, lamb in zip(betas, lambs):
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points_num.append(ell.ell2cart(beta, lamb))
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