Determining surfaces of revolution from their implicit equations

TL;DR

This paper presents an efficient algorithm to determine if an algebraic surface given implicitly is a surface of revolution, and if so, to find its axis and generatrix, also exploring rationality properties of such surfaces.

Contribution

The authors develop a simple, effective method to identify surfaces of revolution from implicit equations and to compute their defining features, addressing a previously unresolved problem.

Findings

Algorithm accurately detects surfaces of revolution from polynomial equations.

Method computes the axis and generatrix when the surface is rotational.

Analysis of rationality and unirationality of surfaces of revolution.

Abstract

Results of number of geometric operations (often used in technical practise, as e.g. the operation of blending) are in many cases surfaces described implicitly. Then it is a challenging task to recognize the type of the obtained surface, find its characteristics and for the rational surfaces compute also their parameterizations. In this contribution we will focus on surfaces of revolution. These objects, widely used in geometric modelling, are generated by rotating a generatrix around a given axis. If the generatrix is an algebraic curve then so is also the resulting surface, described uniquely by a polynomial which can be found by some well-established implicitation technique. However, starting from a polynomial it is not known how to decide if the corresponding algebraic surface is rotational or not. Motivated by this, our goal is to formulate a simple and efficient algorithm whose input is a polynomial with the coefficients from some subfield of $\mathbb{R}$ and the output is the answer whether the shape is a surface of revolution. In the affirmative case we also find the equations of its axis and generatrix. Furthermore, we investigate the problem of rationality and unirationality of surfaces of revolution and show that this question can be efficiently answered discussing the rationality of a certain associated planar curve.