Chapter One - How to Draw a Sphere Part I, Basic Math: January 1995

Abstract

This chapter presents the matrix mathematics that is necessary to manipulate (and ultimately render) second-order surfaces. The chapter describes a special-purpose program that is highly optimized for drawing spheres. There are quite a variety of interesting tricks involved in the special-purpose program. The whole bunch of matrix mathematics is necessary to understand the algorithm. The description and manipulation of second-order curves and surfaces is a very nice example of the use of homogeneous coordinates and projective geometry. The chapter indicates the capability of homogeneous coordinate geometry to deal with points and planes at infinity that are expected to be useful. The focus of attention in making a rendering algorithm fast is the inner loop. Inside this loop there is a requirement of using a normal vector at each pixel, for illumination calculations and texture coordinates , and for texture mapping. With the appropriate initialization one can calculate many of these quantifies incrementally. That is called as a square root, two scalar additions, two vector-scalar multiplications, and four vector additions.