Tri-planar
Forget it: this is not a projection algorithm (an algorithm that gives you UV coordinates), and you cannot get UV coordinates from it. This is a rendering algorithm that gives you the color obtained by mixing the color that you would get with each XYZ planar projection separately.
Cylindrical, spherical
Like planar ones, these are very simple projection algorithms that give you the UV value directly from the XYZ value, without taking into account connectivity with other vertices.
- For cylindrical: convert (x, y, z) to Cylindrical coordinates (ρ, φ, z) and use u = φ and v = z as UV coordinates
- For spherical: convert (x, y, z) to Spherical coordinates (r, θ, φ) and use u = θ and v = φ as UV coordinates
Of course, you can switch the roles of X, Y and Z to the project using a different axis or do some translation / rotation / scaling to have more control (just like you can control the size and orientation of the plane you use for planar projection) .
Cubic
First, you need to determine which “projection face" you assign to each face of your grid. I call the projection faces X, -X, Y, -Y, Z, and -Z, as in the figure below (where I assume that the X, Y, and Z axes are Red, Green, and Blue, respectively):
To do this, you simply discover which normal coordinate (nx, ny, nz) has the largest absolute value and assigns its face to this axis and sign. For instance:
- if n = (0.8, 0.5, 0.3), then the corresponding face X (| nx | is the largest and nx is positive)
- if n = (0.3, 0.8, 0.5), then the corresponding face Y (| ny | is the largest and ny is positive)
- if n = (0.3, -0.8, 0.5), then the corresponding face is -Y (| ny | is the largest and ny is negative)
Then, as soon as you know which side of the projection you assign each face of your grid to, you can apply the corresponding planar projection to the vertices around this face to get a temporary value (u_temp, v_temp) ∈ [0,1] x [0,1 ].
The next step is to convert this value uv_temp ∈ [0,1] x [0,1] to the uv value included in the smaller square, as shown in image A above. For example, if you applied the projection "X", then you want to get uv ∈ [2/3, 3/3] x [2/4, 3/4], then you will do:
u = 2./3. + u_temp/3.; v = 2./4. + v_temp/4.;
Finally, the last step is to remember to duplicate the UV vertices that belong to two faces with different flat projections (borders between different colors in the image). Indeed, some grid vertices can (and should in most cases) be split into several positions in the UV map to give decent results.