Mathematical Preliminaries

Overview

Mathematical operations on points, vectors and matrices are needed for processing information related to geometrical objects. Even in the modelling of a simple three-dimensional scene, vectors and matrices play an important role in specifying an object’s position, orientation and transformations. Methods for lighting, intersection testing, projections, etc., use a series of vector operations. This chapter gives an overview of computations using geometrical primitives and shapes that form the basis for several algorithms presented in subsequent chapters of the book.

Parametric representations are often used in methods involving geometrical primitives. This chapter deals with analytical equations of lines, planes and curves, and their applications in geometrical computations. Properties of three-dimensional transformations are discussed using their matrix representations. The chapter also introduces concepts such as signed area and distance, affine combinations of points and barycentric coordinates.