Finite representations of real parametric curves and surfaces

Abstract

Algebraic curves and surfaces are commonly used in geometric modeling. Parametric curves and surfaces are those that can be represented using rational parametric equations, and are particularly important. In geometric modeling applications, the parametric equations are restricted to some bounded portion of the domain, yielding a segment of a curve or a patch of a surface. However, the algebraic curve or surface is an image of the entire infinite parameter domain. Attempting to map the entire curve or surface using very large regions of the parameter domain is not a solution because some finite points may be images of infinite parameter values.