## Thursday, May 12, 2016

### 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!

#### 1 comment:

Jared M Johnson said...

This is what we use for panos at Google. Seems wasteful vs an equal area projection like Lambert, which is over 50 percent shorter, so more distortion vertically... Would be interesting to store this as interrupted sinusoidal equal-area projection but be able to do the indexing this easily.