﻿ Parametric curves and surfaces

Parametric curves and surfaces
Two main approaches for representing curves and surfaces exist, algebraic/implicit representation and parametric representation. The two approaches have different properties, and for many geometry problems the switching between the two representations make very efficient algorithms. However, for curves and surface having total algebraic degree higher than two switching between the two representations is non-trival. Going from parametric to implicit representation is denoted implicitization, and going from implicit to parametric representation is denoted parametrization.

## Parametric representation

A typical parametric curve in CAD is the mapping of an interval on the real line into two dimensional real space: p:[a,b] → R2. While a tensor product parametric surface map a region into three dimensional real space  r:[a,b]x[c,d]  →  R3:

• p(s)=(x(s),y(s)), where both x(s) and y(s) are polynomial or rational functions of degree n>0.
• r(u,v)=(x(u,v),y(u,v),z(u,v)), where x(u,v),y(u,v) and z(u,v) are polynomial or rational functions of degrees less than (n1, n2), with n1, n2>0.

Published May 25, 2005