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# cone-vector.py --- A Python implementation of cone sampling
# Copyright © 2021 Arun I <arunisaac@systemreboot.net>
# Copyright © 2021 Murugesan Venkatapathi <murugesh@iisc.ac.in>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful, but
# WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU
# General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see
# <https://www.gnu.org/licenses/>.
from numpy import arcsin, cos, dot, empty, ones, sin, sqrt, pi, where, zeros
from numpy.random import randn, random
from numpy.linalg import norm
from scipy.special import betainc, betaincinv, gamma
def random_vector_on_sphere(dim):
"""Return a random vector uniformly distributed on the unit sphere."""
x = randn(dim)
return x / norm(x)
def planar_angle2solid_angle_fraction(planar_angle, dim):
"""Return the solid angle fraction for a given planar angle."""
alpha = (dim - 1) / 2
beta = 1/2
return where(planar_angle < pi/2,
0.5*betainc(alpha, beta, sin(planar_angle)**2),
1 - 0.5*betainc(alpha, beta, sin(planar_angle)**2))
def solid_angle_fraction2planar_angle(solid_angle_fraction, dim):
"""Return the planar angle for a given solid angle fraction."""
alpha = (dim - 1) / 2
beta = 1/2
return where(solid_angle_fraction < 1/2,
arcsin(sqrt(betaincinv(alpha, beta, 2*solid_angle_fraction))),
pi - arcsin(sqrt(betaincinv(alpha, beta, 2*(1-solid_angle_fraction)))))
def rotate_from_nth_canonical(x, axis):
"""Rotate vector from around the nth canonical axis to the given axis.
"""
xn = x[-1]
axisn = axis[-1]
if axisn != 1:
b = norm(axis[:-1])
a = (dot(x, axis) - xn*axisn) / b
s = sqrt(1 - axisn**2)
x = x + (xn*s + a*(axisn - 1))/b * axis
x[-1] = x[-1] + xn*(axisn - 1) - a*s \
- axisn*(xn*s + a*(axisn - 1))/b
return x
def random_vector_on_spherical_cap(axis, maximum_planar_angle):
"""Return a random vector uniformly distributed on a spherical cap."""
dim = axis.size
maximum_solid_angle_fraction = planar_angle2solid_angle_fraction(maximum_planar_angle, dim)
solid_angle_fraction = maximum_solid_angle_fraction*random()
planar_angle = solid_angle_fraction2planar_angle(solid_angle_fraction, dim)
x = empty(dim)
x[:-1] = sin(planar_angle) * random_vector_on_sphere(dim - 1)
x[-1] = cos(planar_angle)
return rotate_from_nth_canonical(x, axis)
# Sample code exercising the above functions
dim = 100
maximum_planar_angle = pi/3
axis = ones(dim)
axis = axis/norm(axis)
print(random_vector_on_spherical_cap(axis, maximum_planar_angle))
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