Determination and (re)parametrization of rational developable surfaces

TL;DR

This paper investigates rational developable surfaces, providing methods to determine developability and to compute or reparametrize rational representations, which are crucial in CAD and geometric modeling.

Contribution

It introduces algorithms for identifying and reparametrizing rational developable surfaces from algebraic or parametric forms.

Findings

Determines developability via geometric inspection.

Provides rational proper parametrizations for developable surfaces.

Offers reparametrization techniques for rational parametric surfaces.

Abstract

The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the stan- dard parametric form, but it can also be in the implicit form which is commonly used in algebraic geometry. Not all algebraic developable surfaces have rational parametrizations. In this paper, we focus on the rational developable surfaces. For a given algebraic surface, we first determine whether it is developable by geometric inspection, and we give a rational proper parametrization for the af- firmative case. For a rational parametric surface, we can also determine the developability and give a proper reparametrization for the developable surface.