spherical to cartesian coords

This is probably easy to google if I had used the right key-words.   Apparently I didn't.   I will derive it here for my own future use.

One of the three formulas I remember learning in the dark ages:

x = rho cos(phi) sin(theta)
y = rho sin(phi) sin(theta)
z = rho cos (theta)

We know this from geometry but we could also square everything and sum it to get:

rho = sqrt(x^2 + y^2 + z^2)

This lets us solve for theta pretty easily:

cos(theta) = z / sqrt(x^2 + y^2 + z^2)

Because sin^2 + cos^2 = 1 we can get:

sin(theta) = sqrt(1 - z^2/( x^2 + y^2 + z^2))

phi we can also get from geometry using the ever useful atan2:

phi = atan2(y, x)