Published
Edited
Mar 20, 2021
3 stars
// Pick a random point uniformly on the cylinder and project
// horizontally onto the sphere. Needs 2 uniform variates.
// Proof this preserves area: Archimedes (225 BCE)
// http://en.wikipedia.org/wiki/On_the_Sphere_and_Cylinder
cylinder_random = () => {
const
a = 2*Math.PI*random(),
u = random(),
s = Math.sqrt(u/(1 - u));
return [Math.cos(a)*s, Math.sin(a)*s];
}
// Marsaglia (1972) http://doi.org/10.1214/aoms/1177692644
// Take a random point in a unit-radius disk and project to
// the 2-sphere via a (2d) radially symmetric function.
// Needs ~2.5 uniform variates.
ball2_rejection_random = () => {
let x, y, r2; do {
x = random() - 0.5;
y = random() - 0.5;
r2 = x*x + y*y;
} while (r2 >= 0.25);
const q = Math.sqrt(0.25 - r2);
return [x/q, y/q];
}
// Normalize magnitude of a point in the unit-radius ball.
// Needs ~5.7 uniform variates.
ball3_rejection_random = () => {
let x, y, z, r; do {
x = random() - 0.5;
y = random() - 0.5;
z = random() - 0.5;
r = Math.hypot(x, y, z);
} while (r >= 0.25);
const q = z + r;
if (q === 0) return [Infinity, 0];
return [x/q, y/q];
}
// Cook (1957) https://doi.org/10.1090/S0025-5718-1957-0690630-7
// This needs ~13.0 uniform variates per point, but has the
// advantage of requiring no square roots or circular functions,
// only rational arithmetic.
ball4_rejection_random = () => {
let a, b, c, d; do {
a = random() - 0.5;
b = random() - 0.5;
c = random() - 0.5;
d = random() - 0.5;
} while (a*a + b*b + c*c + d*d >= 0.25);
const q = a*a + b*b;
if (q === 0) return [Infinity, 0];
return [(a*c + b*d)/q, (a*d - b*c)/q];
}
// Normalize 3-tuple of gaussian variates.
gauss3_random = () => {
const
x = randn(),
y = randn(),
z = randn(),
q = z + Math.hypot(x, y, z);
if (q === 0) return [Infinity, 0];
return [x/q, y/q];
}
// Like `ball4_rejection_random` but using gaussian variates.
// Any radially symmetric distribution in 4-space would work.
gauss4_random = () => {
const
a = randn(),
b = randn(),
c = randn(),
d = randn(),
q = a*a + b*b;
if (q === 0) return [Infinity, 0];
return [(a*c + b*d)/q, (a*d - b*c)/q];
}
// Inverse project stereographic to geographic coordinates.
lonlat = ([x, y]) => [
180/Math.PI * Math.atan2(y, x),
90 - 360/Math.PI * Math.atan(Math.hypot(x, y))
]
// Any infinite coordinate represents the south pole.
lonlat([Infinity, 0])
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