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