Stereo Projections: Background, Mathematics, and Use

Perspective Projections

We will begin by deriving the basic perspective projection matrix, assuming geometry is stored in the eye coordinate system (i.e., the eye point, E is at the origin. Then we will see how to generate stereo projections by applying left and right eye shears to this matrix. That derivation will be accomplished by explicitly positioning the eye points at (±d/2,0,0) as indicated earlier in these notes. The interested reader can compare these results with those for a completely general placement of the eye by studyinging section VII of the previously mentioned technical report on the Mathematics of Graphical Transformations.

Simple Perspective Projection

As the image on this page indicates, an arbitrary point, P is projected towards the eye to P′. We are given the coordinates of P, and we assume the projection plane is z=z_pp. From the picture, it is clear that the two unknowns can be computed based on a similar triangle (or constant slope) argument as follows:

That is,

and