Determination and (re)parametrization of rational developable surfaces

Abstract

The developable surface is an important surface in computer aided design, geometric modeling and industrial manufactory. It is often given in the standard 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 affirmative case. For a rational parametric surface, we can also determine the developability and give a proper reparametrization for the developable surface.

Detecting When an Implicit Equation or a Rational Parametrization Defines a Conical or Cylindrical Surface, or a Surface of Revolution

“…In this section we compare our results with the results in [5], [9], [7], [12], [1] and [16], where similar problems to ours, or somehow connected to ours, are addressed.…”

“…Recognizing cylindrical surfaces and surfaces of revolution is also addressed in [11], [12], and the results in these papers are extended in [7] to conical surfaces and other types of surfaces not treated in this paper, like helical surfaces and spiral surfaces. Recognizing surfaces of revolution is also investigated in [1], [16] and some aspects regarding the recognition of cylindrical and conical surfaces are studied in [9]. In Section 4 of this paper, we compare our results with the results in these papers.…”

“…In [9] the authors treat a different problem, namely recognizing and (re)parametrizing rational developable surfaces. However, the problem of identifying cylindrical and conical surfaces, both in implicit and in rational form, also comes up in this paper.…”

“…However, the problem of identifying cylindrical and conical surfaces, both in implicit and in rational form, also comes up in this paper. First, the results in [9] require recognizing that the surface is developable. This is done by checking whether certain determinants depending on some derivatives of the polynomial implicitly defining the surface (see Theorem 3.1 in [9]) or the parametrization of the surface (see Theorem 4.1 in [9]) are identically zero.…”

“…First, the results in [9] require recognizing that the surface is developable. This is done by checking whether certain determinants depending on some derivatives of the polynomial implicitly defining the surface (see Theorem 3.1 in [9]) or the parametrization of the surface (see Theorem 4.1 in [9]) are identically zero. Once the surface has been identified as developable, algorithms for detecting whether the surface is cylindrical or conical are provided.…”