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curve3d.py
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curve3d.py
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'''
Describe 3D curve by segments;
each segment described by theta and psi (angle in yz plane):
x_{k+1} = x_k + ds * |sin(theta)| * cos(psi)
y_{k+1} = y_k + ds * |sin(theta)| * sin(psi)
z_{k+1} = z_k + ds * cos(theta)
Taking only absolute values of sine allows us to consider theta from 0 to 2 pi.
NB: It's safe to take psi % (2* pi), but theta % pi is not correct - for that there is a function defined below
'''
import scipy as sp
from math import cos, sin, acos, atan2, pi
def calc_slist(xx, yy=None, zz=None):
if yy is None:
yy = sp.zeros_like(xx)
if zz is None:
zz = sp.zeros_like(xx)
slist = sp.zeros_like(xx, dtype=sp.float64)
sp.cumsum((sp.diff(xx) ** 2 + sp.diff(yy) ** 2 + sp.diff(zz) ** 2) ** (1 / 2), out=slist[1:])
return slist
def calc_length(xx, yy, zz):
return sp.sum((sp.diff(xx) ** 2 + sp.diff(yy) ** 2 + sp.diff(zz) ** 2) ** (1 / 2))
def get_ds_angles(dx, dy, dz):
ds = (dx ** 2 + dy ** 2 + dz ** 2) ** (1 / 2)
theta = acos(dz / ds) # From 0 to pi
psi = atan2(dy, dx) % (2 * pi) # From 0 to 2pi
return ds, theta, psi
def get_dr(ds, theta, psi):
dx = ds * sin(theta) * cos(psi)
dy = ds * sin(theta) * sin(psi)
dz = ds * cos(theta)
return dx, dy, dz
def build_curve(r0, ds, thetas, psis):
'''
:param r0: start position
:param thetas:
:param psis:
:param ds: constant size of segment
:return: rr: sp.array([x0,y0,z0],...])
'''
if len(thetas) != len(psis):
raise ValueError
r = sp.array(r0, dtype=float)
rr = [r]
for theta, psi in zip(thetas, psis):
dr = get_dr(ds, theta, psi)
r = r + dr
rr.append(r)
rr = sp.array(rr)
return rr
def reconstruct_angles(rr, num_segments):
'''
'''
from scipy.interpolate import pchip_interpolate
r0 = rr[0]
xx, yy, zz = rr.T
slist = calc_slist(xx, yy, zz)
slist_norm = sp.linspace(0, 1, num_segments + 1, endpoint=True)
# Interpolate to equal arc length spacing
xx_interp = pchip_interpolate(slist / slist[-1], xx, slist_norm)
yy_interp = pchip_interpolate(slist / slist[-1], yy, slist_norm)
zz_interp = pchip_interpolate(slist / slist[-1], zz, slist_norm)
thetas = []
psis = []
for dx, dy, dz in zip(sp.diff(xx_interp), sp.diff(yy_interp), sp.diff(zz_interp)):
_, theta, psi = get_ds_angles(dx, dy, dz)
thetas.append(theta)
psis.append(psi)
ds = slist[-1] / num_segments
return r0, ds, thetas, psis
def _theta_to_pi_interval(theta):
theta = theta % (2 * pi)
if theta >= pi:
theta = 2 * pi - theta # mirror
return theta
thetas_to_pi_interval = sp.vectorize(_theta_to_pi_interval)
## Pairwise operations
def calc_cross_from_angles(ds1, theta1, psi1, ds2, theta2, psi2):
'''
Return euclidian form of dr1 x dr2, given angles defining dr1 and dr2.
'''
dr_cross = sp.array([cos(theta2) * sin(psi1) * sin(theta1) - cos(theta1) * sin(psi2) * sin(theta2),
-(cos(psi1) * cos(theta2) * sin(theta1)) + cos(psi2) * cos(theta1) * sin(theta2),
-(sin(psi1 - psi2) * sin(theta1) * sin(theta2))])
return ds1 * ds2 * dr_cross
# # Test reconstruct angles
# if __name__ == '__main__':
# # OK: get_ds_angles
# # OK: reconstructed angles on spiral; N=M
# # NOT OK: reconstructed angles, when interpolated - hypothesis - interpolation is not precise; redo interpolation
#
# import numpy.random as rnd
# import quick
#
# dx, dy, dz = 3, 0, 0
# # print(get_ds_angles(dx,dy,dz))
# res = get_ds_angles(dx, dy, dz)
# print(get_dr(*res))
#
# dx, dy, dz = 0, 5, 0
# res = get_ds_angles(dx, dy, dz)
# print(get_dr(*res))
#
# dx, dy, dz = 0, 0, 6
# res = get_ds_angles(dx, dy, dz)
# print(get_dr(*res))
#
# dx, dy, dz = 0, 0, -6
# res = get_ds_angles(dx, dy, dz)
# print(get_dr(*res))
#
# dx, dy, dz = 1, 1, 1
# res = get_ds_angles(dx, dy, dz)
# print(get_dr(*res))
#
# dx, dy, dz = -1, 10, 121
# res = get_ds_angles(dx, dy, dz)
# print(get_dr(*res))
#
# # # Spiral
# N = 50
# M = 40
# r0 = (1, 0, 0)
# ds = 1
# thetas = [sp.arccos(0.5) for _ in range(N)]
# psis = [2 * sp.pi / N * k for k in range(N)]
# rr = build_curve(r0, ds, thetas, psis)
#
# r0_rec, ds_rec, thetas_rec, psis_rec = reconstruct_angles(rr, M)
#
# with quick.Plot() as qp:
# qp.plot(sp.linspace(0, 1, N, endpoint=True), thetas)
# qp.plot(sp.linspace(0, 1, M, endpoint=True), thetas_rec, 'o')
# qp.ylabel(r"$\theta$")
#
# with quick.Plot() as qp:
# qp.plot(sp.linspace(0, 1, N, endpoint=True), psis)
# qp.plot(sp.linspace(0, 1, M, endpoint=True), psis_rec, 'o')
# qp.ylabel(r"$\psi$")
# ## Test cross product
# if __name__ == '__main__':
# # OK: orthogonal vectors
# # OK: right-hand triplet
# # OK: Anti-symmetry
# # OK: Length scaling
# # OK: zero on parallel vectors
# # OK: non-orthogonal, non-parallel
#
# r1 = (1, 0, 1)
# r2 = (0.5, 0.5, 0)
#
# angles1 = get_ds_angles(*r1)
# angles2 = get_ds_angles(*r2)
#
# print(calc_cross_from_angles(*angles1, *angles2))
# print(calc_cross_from_angles(*angles2, *angles1))
# ## Test thetas to pi interval
# if __name__ == '__main__':
# # OK: preserves cosine
# # OK: maps to [0,pi)
# # OK: works on 2D arrays
# import quick
# thetas = sp.linspace(- 2 * pi, 4 * pi, 100, endpoint=True)
#
# thetas_new = thetas_to_pi_interval(thetas)
# with quick.Plot() as qp:
# qp.plot(thetas, sp.cos(thetas))
# qp.plot(thetas, sp.cos(thetas_new), 'ro')
# with quick.Plot() as qp:
# qp.plot(thetas, thetas_new)