Curvature-dependent parameterization of curves and surfaces

Abstract

An efficient way of drawing parametric curves and surfaces is to approximate the curve or surface by a sequence of straight-line segments or a mesh of polygons, respectively. In such an approximation, many small line segments or polygons are needed in regions of high curvature, and fewer and larger ones are needed in regions of low curvature. In the paper, special parameterizations of curves and surfaces, called angle parametrizations, are defined such that good approximations to curves and surfaces can be found by uniformly subdividing the parameter space, and applying the special parameterization. A pleasing feature of these parameterizations is that they are easily computed numerically. The numerical methods only require explicit expressions for the 1st- and 2nd-order derivatives of the original parameterization. This implies that the methods presented are applicable to the polynomial and rational curves and surfaces customarily used in computer-aided design, but are not limited to these. Some specific examples are presented. The numerical algorithm for surfaces is inherently suited to implementation on a network of processors operating in parallel, and such an implementation is briefly discussed.