192 lines
7.4 KiB
Python
192 lines
7.4 KiB
Python
import numpy as np
|
|
import ellipsoide
|
|
import Numerische_Integration.num_int_runge_kutta as rk
|
|
import winkelumrechnungen as wu
|
|
import ausgaben as aus
|
|
import GHA.rk as ghark
|
|
from scipy.special import factorial as fact
|
|
from math import comb
|
|
|
|
# Panou, Korakitits 2019
|
|
|
|
def p_q(ell: ellipsoide.EllipsoidTriaxial, x, y, z):
|
|
H = x ** 2 + y ** 2 / (1 - ell.ee ** 2) ** 2 + z ** 2 / (1 - ell.ex ** 2) ** 2
|
|
|
|
n = np.array([x / np.sqrt(H), y / ((1 - ell.ee ** 2) * np.sqrt(H)), z / ((1 - ell.ex ** 2) * np.sqrt(H))])
|
|
|
|
beta, lamb, u = ell.cart2ell(x, y, z)
|
|
carts = ell.ell2cart(beta, lamb, u)
|
|
B = ell.Ex ** 2 * np.cos(beta) ** 2 + ell.Ee ** 2 * np.sin(beta) ** 2
|
|
L = ell.Ex ** 2 - ell.Ee ** 2 * np.cos(lamb) ** 2
|
|
|
|
c1 = x ** 2 + y ** 2 + z ** 2 - (ell.ax ** 2 + ell.ay ** 2 + ell.b ** 2)
|
|
c0 = (ell.ax ** 2 * ell.ay ** 2 + ell.ax ** 2 * ell.b ** 2 + ell.ay ** 2 * ell.b ** 2 -
|
|
(ell.ay ** 2 + ell.b ** 2) * x ** 2 - (ell.ax ** 2 + ell.b ** 2) * y ** 2 - (
|
|
ell.ax ** 2 + ell.ay ** 2) * z ** 2)
|
|
t2 = (-c1 + np.sqrt(c1**2 - 4*c0)) / 2
|
|
t1 = c0 / t2
|
|
t2e = ell.ax ** 2 * np.sin(lamb) ** 2 + ell.ay ** 2 * np.cos(lamb) ** 2
|
|
t1e = ell.ay ** 2 * np.sin(beta) ** 2 + ell.b ** 2 * np.cos(beta) ** 2
|
|
|
|
F = ell.Ey ** 2 * np.cos(beta) ** 2 + ell.Ee ** 2 * np.sin(lamb) ** 2
|
|
p1 = -np.sqrt(L / (F * t2)) * ell.ax / ell.Ex * np.sqrt(B) * np.sin(lamb)
|
|
p2 = np.sqrt(L / (F * t2)) * ell.ay * np.cos(beta) * np.cos(lamb)
|
|
p3 = 1 / np.sqrt(F * t2) * (ell.b * ell.Ee ** 2) / (2 * ell.Ex) * np.sin(beta) * np.sin(2 * lamb)
|
|
# p1 = -np.sign(y) * np.sqrt(L / (F * t2)) * ell.ax / (ell.Ex * ell.Ee) * np.sqrt(B) * np.sqrt(t2 - ell.ay ** 2)
|
|
# p2 = np.sign(x) * np.sqrt(L / (F * t2)) * ell.ay / (ell.Ey * ell.Ee) * np.sqrt((ell.ay ** 2 - t1) * (ell.ax ** 2 - t2))
|
|
# p3 = np.sign(x) * np.sign(y) * np.sign(z) * 1 / np.sqrt(F * t2) * ell.b / (ell.Ex * ell.Ey) * np.sqrt(
|
|
# (t1 - ell.b ** 2) * (t2 - ell.ay ** 2) * (ell.ax ** 2 - t2))
|
|
p = np.array([p1, p2, p3])
|
|
q = np.array([n[1]*p[2]-n[2]*p[1],
|
|
n[2]*p[0]-n[0]*p[2],
|
|
n[1]*p[1]-n[1]*p[0]])
|
|
|
|
return {"H": H, "n": n, "beta": beta, "lamb": lamb, "u": u, "B": B, "L": L, "c1": c1, "c0": c0, "t1": t1, "t2": t2,
|
|
"F": F, "p": p, "q": q}
|
|
def gha1_num(ell: ellipsoide.EllipsoidTriaxial, x, y, z, alpha0, s, num):
|
|
values = p_q(ell, x, y, z)
|
|
H = values["H"]
|
|
p = values["p"]
|
|
q = values["q"]
|
|
|
|
dxds0 = p[0] * np.sin(alpha0) + q[0] * np.cos(alpha0)
|
|
dyds0 = p[1] * np.sin(alpha0) + q[1] * np.cos(alpha0)
|
|
dzds0 = p[2] * np.sin(alpha0) + q[2] * np.cos(alpha0)
|
|
|
|
h = lambda dxds, dyds, dzds: dxds**2 + 1/(1-ell.ee**2)*dyds**2 + 1/(1-ell.ex**2)*dzds**2
|
|
|
|
f1 = lambda s, x, dxds, y, dyds, z, dzds: dxds
|
|
f2 = lambda s, x, dxds, y, dyds, z, dzds: -h(dxds, dyds, dzds) / H * x
|
|
f3 = lambda s, x, dxds, y, dyds, z, dzds: dyds
|
|
f4 = lambda s, x, dxds, y, dyds, z, dzds: -h(dxds, dyds, dzds) / H * y/(1-ell.ee**2)
|
|
f5 = lambda s, x, dxds, y, dyds, z, dzds: dzds
|
|
f6 = lambda s, x, dxds, y, dyds, z, dzds: -h(dxds, dyds, dzds) / H * z/(1-ell.ex**2)
|
|
|
|
funktionswerte = rk.verfahren([f1, f2, f3, f4, f5, f6], [0, x, dxds0, y, dyds0, z, dzds0], s, num)
|
|
return funktionswerte
|
|
|
|
def checkLiouville(ell: ellipsoide.EllipsoidTriaxial, points):
|
|
constantValues = []
|
|
for point in points:
|
|
x = point[1]
|
|
dxds = point[2]
|
|
y = point[3]
|
|
dyds = point[4]
|
|
z = point[5]
|
|
dzds = point[6]
|
|
|
|
values = p_q(ell, x, y, z)
|
|
p = values["p"]
|
|
q = values["q"]
|
|
t1 = values["t1"]
|
|
t2 = values["t2"]
|
|
|
|
P = p[0]*dxds + p[1]*dyds + p[2]*dzds
|
|
Q = q[0]*dxds + q[1]*dyds + q[2]*dzds
|
|
alpha = np.arctan(P/Q)
|
|
|
|
c = ell.ay**2 - (t1 * np.sin(alpha)**2 + t2 * np.cos(alpha)**2)
|
|
constantValues.append(c)
|
|
pass
|
|
|
|
|
|
def gha1_ana(ell: ellipsoide.EllipsoidTriaxial, x, y, z, alpha0, s, maxM):
|
|
"""
|
|
Panou, Korakitits 2020, 5ff.
|
|
:param ell:
|
|
:param x:
|
|
:param y:
|
|
:param z:
|
|
:param alpha0:
|
|
:param s:
|
|
:param maxM:
|
|
:return:
|
|
"""
|
|
x_m = [x]
|
|
y_m = [y]
|
|
z_m = [z]
|
|
|
|
# erste Ableitungen (7-8)
|
|
sqrtH = np.sqrt(ell.H(x, y, z))
|
|
n = np.array([x / sqrtH,
|
|
y / ((1-ell.ee**2) * sqrtH),
|
|
z / ((1-ell.ex**2) * sqrtH)])
|
|
u, v = ell.cart2para(x, y, z)
|
|
G = np.sqrt(1 - ell.ex**2 * np.cos(u)**2 - ell.ee**2 * np.sin(u)**2 * np.sin(v)**2)
|
|
q = np.array([-1/G * np.sin(u) * np.cos(v),
|
|
-1/G * np.sqrt(1-ell.ee**2) * np.sin(u) * np.sin(v),
|
|
1/G * np.sqrt(1-ell.ex**2) * np.cos(u)])
|
|
p = np.array([q[1]*n[2] - q[2]*n[1],
|
|
q[2]*n[0] - q[0]*n[2],
|
|
q[1]*n[1] - q[1]*n[0]])
|
|
x_m.append(p[0] * np.sin(alpha0) + q[0] * np.cos(alpha0))
|
|
y_m.append(p[1] * np.sin(alpha0) + q[1] * np.cos(alpha0))
|
|
z_m.append(p[2] * np.sin(alpha0) + q[2] * np.cos(alpha0))
|
|
|
|
# H Ableitungen (7)
|
|
H_ = lambda p: np.sum([comb(p, i) * (x_m[p - i] * x_m[i] +
|
|
1 / (1-ell.ee**2) ** 2 * y_m[p-i] * y_m[i] +
|
|
1 / (1-ell.ex**2) ** 2 * z_m[p-i] * z_m[i]) for i in range(0, p+1)])
|
|
|
|
# h Ableitungen (7)
|
|
h_ = lambda q: np.sum([comb(q, j) * (x_m[q-j+1] * x_m[j+1] +
|
|
1 / (1 - ell.ee ** 2) ** 2 * y_m[q-j+1] * y_m[j+1] +
|
|
1 / (1 - ell.ex ** 2) ** 2 * z_m[q-j+1] * z_m[j+1]) for j in range(0, q + 1)])
|
|
|
|
# h/H Ableitungen (6)
|
|
hH_ = lambda t: 1/H_(0) * (h_(t) - fact(t) *
|
|
np.sum([H_(t+1-l)/(fact(t+1-l)*fact(l-1))*hH_t[l-1] for l in range(1, t+1)]))
|
|
|
|
# xm, ym, zm Ableitungen (6)
|
|
x_ = lambda m: -np.sum([comb(m-2, k) * hH_t[m-2-k] * x_m[k] for k in range(0, m-2+1)])
|
|
y_ = lambda m: -1/(1-ell.ee**2) * np.sum([comb(m-2, k) * hH_t[m-2-k] * y_m[k] for k in range(0, m-2+1)])
|
|
z_ = lambda m: -1/(1-ell.ex**2) * np.sum([comb(m-2, k) * hH_t[m-2-k] * z_m[k] for k in range(0, m-2+1)])
|
|
|
|
hH_t = []
|
|
a_m = []
|
|
b_m = []
|
|
c_m = []
|
|
for m in range(2, maxM+1):
|
|
hH_t.append(hH_(m-2))
|
|
x_m.append(x_(m))
|
|
a_m.append(x_m[m] / fact(m))
|
|
y_m.append(y_(m))
|
|
b_m.append(y_m[m] / fact(m))
|
|
z_m.append(z_(m))
|
|
c_m.append(z_m[m] / fact(m))
|
|
|
|
# am, bm, cm (6)
|
|
x_s = 0
|
|
for a in a_m:
|
|
x_s = x_s * s + a
|
|
y_s = 0
|
|
for b in b_m:
|
|
y_s = y_s * s + b
|
|
z_s = 0
|
|
for c in c_m:
|
|
z_s = z_s * s + c
|
|
|
|
return x_s, y_s, z_s
|
|
pass
|
|
|
|
|
|
if __name__ == "__main__":
|
|
ell = ellipsoide.EllipsoidTriaxial.init_name("Eitschberger1978")
|
|
ellbi = ellipsoide.EllipsoidTriaxial.init_name("Bessel-biaxial")
|
|
re = ellipsoide.EllipsoidBiaxial.init_name("Bessel")
|
|
x0 = 5672455.1954766
|
|
y0 = 2698193.7242382686
|
|
z0 = 1103177.6450055107
|
|
alpha0 = wu.gms2rad([20, 0, 0])
|
|
s = 100
|
|
num = 100
|
|
werteTri = gha1_num(ellbi, x0, y0, z0, alpha0, s, num)
|
|
print(aus.xyz(werteTri[-1][1], werteTri[-1][3], werteTri[-1][5], 8))
|
|
print(np.sqrt((x0-werteTri[-1][1])**2+(y0-werteTri[-1][3])**2+(z0-werteTri[-1][5])**2))
|
|
checkLiouville(ell, werteTri)
|
|
werteBi = ghark.gha1(re, x0, y0, z0, alpha0, s, num)
|
|
print(aus.xyz(werteBi[0], werteBi[1], werteBi[2], 8))
|
|
print(np.sqrt((x0-werteBi[0])**2+(y0-werteBi[1])**2+(z0-werteBi[2])**2))
|
|
|
|
werteAna = gha1_ana(ell, x0, y0, z0, alpha0, s, 7)
|
|
print(aus.xyz(werteAna[0], werteAna[1], werteAna[2], 8)) |