Recognizing implicitly given rational canal surfaces

TL;DR

This paper develops an efficient algorithm to identify and parameterize rational canal surfaces from implicit polynomial representations, facilitating their recognition and modeling in geometric design.

Contribution

It introduces a simple, effective method to determine if an algebraic surface is a rational canal surface and computes its rational parameterization.

Findings

Algorithm successfully identifies rational canal surfaces from implicit equations.

Provides a rational parameterization of the squared medial axis transform.

Enhances geometric modeling and CAD applications with improved recognition techniques.

Abstract

It is still a challenging task of today to recognize the type of a given algebraic surface which is described only by its implicit representation. In~this paper we will investigate in more detail the case of canal surfaces that are often used in geometric modelling, Computer-Aided Design and technical practice (e.g. as blending surfaces smoothly joining two parts with circular ends). It is known that if the squared medial axis transform is a rational curve then so is also the corresponding surface. However, starting from a polynomial it is not known how to decide if the corresponding algebraic surface is rational canal surface or not. Our goal is to formulate a simple and efficient algorithm whose input is a~polynomial with the coefficients from some subfield of real numbers and the output is the answer whether the surface is a rational canal surface. In the affirmative case we also compute a rational parameterization of the squared medial axis transform which can be then used for finding a rational parameterization of the implicitly given canal surface.