Math: Random Points

Some useful functions, for generating uniformly distributed random points
by @_sorceress, March 2014
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- All functions input a radius 'r'.
- 'random' should return a uniformly distributed random number between 0 and 1.

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Point (x,y) on the circle perimeter

a = 2*pi*random

x = r*cos(a)
y = r*sin(a)
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Point (x,y) within the circular disk

a = 2*pi*random
b = sqrt(random)*r

x = b*cos(a)
y = b*sin(a)
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Point (x,y) within the annulus, with internal radius m*r (where 0<m<1)

a = 2*pi*random
b = m*m
b = r*sqrt(b+random*(1-b))

x = b*cos(a)
y = b*sin(a)
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Point (x,y,z) within the cylinder, with height h

a = 2*pi*random
b = sqrt(random)*r

x = b*cos(a)
y = b*sin(a)
z = h*random
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Point (x,y,z) on the spherical surface

a = 2*random-1
b = 2*pi*random
c = r*sqrt(1-a*a)

x = c*cos(b)
y = c*sin(b)
z = r*a
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Point (x,y,z) within the spherical ball

d = r*random^(1/3)
a = 2*random-1
b = 2*pi*random
c = d*sqrt(1-a*a)

x = c*cos(b)
y = c*sin(b)
z = d*a
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Point (x,y,z) within the spherical ball (generally faster, but unpredictable halting time)

do {
x = r*(2*random-1)
y = r*(2*random-1)
z = r*(2*random-1)
} while(x*x+y*y+z*z>r*r)
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Point (x,y,z) within the thick spherical shell, with inner radius m*r (where 0<m<1)

d = m*m*m
d = r*(d+random*(1-d))^(1/3)
a = 2*random-1
b = 2*pi*random
c = d*sqrt(1-a*a)

x = c*cos(b)
y = c*sin(b)
z = d*a
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