Numerical proper reparametrization of space curves and surfaces

Abstract

Simplifying rational parametrizations of surfaces is a basic problem in CAD (computer-aided design). One important way is to reduce their tracing index, called proper reparametrization. Most existing proper reparametrization work is symbolic, yet in practical environments the surfaces are usually given with perturbed coefficients hence need a numerical technique of reparametrization considering the intrinsic properness of the perturbed surfaces. We present algorithms for reparametrizing a numerically rational space curve or surface. First, we provide an efficient way to find a parametric support transformation and compute a reparametrization with proper parametric support. Second, we develop a numerical algorithm to further reduce the tracing index, where numerical techniques such as sparse interpolation and approximated GCD computations are involved. We finally provide the error bound between the given rational curve/surface and our reparametrization result.