equirectangular image to spherical coords

An equirectangular image, popular in 360 video, is a projection that has equal area on the rectangle match area on the sphere.   Here it is for the Earth:

Equirectangular projection (source wikipedia)

This projection is much simpler than I would expect.    The area on the unit radius sphere from theta1 to theta2 (I am using the graphics convention of theta is the angle down from the pole) is:

area = 2*Pi*integral sin(theta) d_theta = 2*Pi*(cos(theta_1) - cos(theta_2))

In Cartesian coordinates this is just:

area = 2*Pi*(z_1 - z_2)

So we can just project the sphere points in the xy plane onto the unit radius cylinder and unwrap it!   If we have such an image with texture coordinates (u,v) in [0,1]^2, then

phi = 2*Pi*u
cos(theta) = 2*v -1

and the inverse:

u = phi / (2*Pi)
v = (1 + cos(theta)) / 2

So yes this projection has singularities at the poles, but it's pretty nice algebraically!