By using one of the two methods shown above, we can effectively drop the Blue channel of our normal map and still derive a proper UV map. This is particularly useful for tiling textures. This can be done because the vector length is known to be one (1). So by using the calculation below we can derive the Blue Channel normal from only a Red and Green channel image, thus freeing the Blue channel up to be used for something else such as Ambient Occlusion or Height for displacement.

b = sqrt(1.0 - dot(rg,rg))

or

b = ((r+r)^2 + (g+g)^2) ^ 1/2

This operation takes the square root of (r^2,g^2) in order to normalize (clamp to 0-1 space) the vector information provided from the math.

Note that this only works in engines like Unreal Engine 4 that will drop the Blue channel on compression and recalculate based on the red and green channels anyway. Otherwise, we would need all 3 channels used for the normal map. Regardless, retaining all 3 channels is considered a good practice because the shader cost of running a square root operation is, arguably, as expensive as a texture channel lookup.