Chapter 1 - Introduction: Foundations

Abstract

This chapter is designed to examine and discuss ambient mathematical spaces. Four different kinds of mathematical spaces support the representation and analysis of free-form curves and surfaces: vector spaces, affine spaces, Grassmann spaces, and projective spaces. A vector space is a collection of objects called vectors that can be added and subtracted as well as multiplied by constants. An affine space is a collection of elements called points for which affine combinations are defined. In Grassmann spaces, points have mass as well as location. Projective spaces introduce points at infinity that are convenient for investigating intersections and poles. This chapter also reviews barycentric coordinates, a topic that is central to the construction of conventional triangular surface patches. It also describes coordinate systems which include rectangular, affine, Grassmann, homogeneous, and barycentric; curve and surface representations which include explicit, implicit, parametric, and procedural. This chapter uses parametric representation for curves and surfaces, and settles upon affine spaces for modeling such polynomial schemes. It adopts a coordinate-free approach for the range, but chooses barycentric coordinates for representing the domain. Grassmann spaces and projective spaces are illustrated to prepare the way for investigating rational curves and surfaces.