Factoring a Homogeneous Transformation for a more Efficient Graphics Pipeline

Abstract

We identify an intermediate coordinate system situated between world coordinates and display coordinates, which exhibits unique features for lighting calculations and for clipping in homogeneous coordinates. Our key contribution is an algorithm for extracting such a coordinate system from a homogeneous viewing transformation that relates WC to DC. The algorithm is based on factoring the transformation into a product of a Euclidean factor and a sparse (computationally cheap) but non- Euclidean factor. A particularly strong application of the proposed technique is the graphical processing of curved surface primitives, such as what is needed in the PHIGS PLUS viewing pipeline. Furthermore, in PHIGS PLUS the graphical data is retained by the graphics system, therefore, it is possible to perform the factoring of the viewing transformation at creation time, and to take advantage of this factored form at traversal time.