1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
|
# sambal --- Sample balls, spheres, spherical caps
# 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 cos, dot, empty, log, sin, sqrt, pi
from numpy.random import randn, random
from numpy.linalg import norm
def random_on_sphere(dim):
"""Return a random vector uniformly distributed on the unit sphere."""
x = randn(dim)
return x / norm(x)
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_on_disk(axis, planar_angle):
"""Return a random vector uniformly distributed on the periphery of a
disk."""
dim = axis.size
x = empty(dim)
x[:-1] = sin(planar_angle) * random_on_sphere(dim - 1)
x[-1] = cos(planar_angle)
return rotate_from_nth_canonical(x, axis)
def random_on_cap(axis, maximum_planar_angle):
"""Return a random vector uniformly distributed on a spherical
cap. The random planar angle is generated using rejection sampling.
"""
# We apply the log function just to prevent the floats from
# underflowing.
dim = axis.size
box_height = (dim-2)*log(sin(min(maximum_planar_angle, pi/2)))
while True:
theta = maximum_planar_angle*random()
f = box_height + log(random())
if f < (dim-2)*log(sin(theta)):
break
return random_on_disk(axis, theta)
|
