How Spherical Coordinates work (no pictures, sorry)

HowSphericalCoordinatesWork.txt

say you have a coordinate system
x axis
y axis
z axis

and a vector v

if you think of the coordinates as a vector (say x) and a plane (say yz)

v makes a half-angle alpha [0,180] with x
v makes a full-angle beta  [0,90] with yz

r is the length of v (r stands for radius)
v.x = rcos(alpha)
v.y = rsin(alpha)cos(beta)
v.z = rsin(alpha)sin(beta)

when alpha == 0, v is aligned with the x axis.
because of this, v.y and v.z need to be 0 when alpha is 0.
In order to make this true, we put sin(alpha) as part of v.y and v.z
because sin(0) = 0

r = sqrt(v.x * v.x + v.y * v.y + v.z * v.z)
alpha = acos(v.x/r)
beta = atan(v.z / v.y)