Stereo Projections: Background, Mathematics, and Use

Towards a Matrix Representation...

The expressions derived on the previous page (x′ = z_ppx/z and y′ = z_ppy/z) are rational linear. That is, they are expressed as a quotient of terms that are linear in the coordinates. A rational linear expression such as this cannot be represented as an affine transformation.

What we can do, however, is to embed our original point P in projective space (obtaining P^P) and then create a 4x4 matrix M_4x4 which, when used to multiply P^P, yields a projective space point which is the projective equivalent of the point we want.

Specifically, the point we want is P′ = (x′, y′, z_pp) = (z_ppx/z, z_ppy/z, z_pp), and a point in projective space that is projectively equivalent to this point is P^P′ = (z_ppx, z_ppy, z_ppz, z). The matrix operation shown above computes the projectively equivalent point. Interactive graphics engines generally create and employ a matrix like this one and then perform the "perspective divide" to obtain the actual desired perspective-mapped point in affine space.

The matrix actually used is generally not exactly the one shown above, however. The next page describes some additional considerations that lead us to use a slightly modified version of this matrix.