Earth Projections

Click the shapes to see the Earth projected onto them using spherical projection mapping:

Usually, textures are mapped to 3D meshes using explicit UV mapping - each vertex is assigned specific UV (texture-space) coordinates. However, UV coordinates can also be calculated procedurally from vertex positions.

Projection mapping is a texture mapping technique that calculates UV coordinates dynamically by mapping between the texture and some intermediate surface, then projecting this surface onto a 3D object.

In the case of spherical projection mapping, we map each vertex to a point on a sphere, ignoring the radius. Given cartesian coordinates (x,y,z)(x,y,z) of a vertex we want to find its spherical coordinates (r,θ,ϕ)(r,\theta,\phi):

r=x2+y2+z2θ=arctan⁡2(y,x)ϕ=arccos⁡(y/r)\begin{aligned} r &= \sqrt{x^2 + y^2 + z^2} \\ \theta &= \arctan2(y, x) \\ \phi &= \arccos(y / r) \end{aligned}

To project through the unit sphere, we can ignore rr and normalize to [0,1][0,1]:

u=0.5+θ2π,v=1−ϕπu = 0.5 + \frac{\theta}{2\pi}, \quad v = 1 - \frac{\phi}{\pi}

You might notice a little seam on one side of the Earth. This is due to the way the uv coordinates are calculated. The problem is described in detail in Distinctive Derivative Differences.

You can find the GLSL source code on github