Missing sets in rational parametrizations of surfaces of revolution

TL;DR

This paper investigates the limitations of rational parametrizations for surfaces of revolution, identifying specific missing regions and providing a simple description of the uncovered parts.

Contribution

It introduces a straightforward method to describe the missing sets in rational parametrizations of surfaces of revolution generated by real rational profile curves.

Findings

The missing set can be described as a union of a circle and a mirror curve.

The analysis applies to surfaces generated by real rational profile curves.

The approach simplifies understanding of parametrization coverage issues.

Abstract

Parametric representations do not cover, in general, the whole geometric object that they parametrize. This can be a problem in practical applications. In this paper we analyze the question for surfaces of revolution generated by real rational profile curves, and we describe a simple small superset of the real zone of the surface not covered by the parametrization. This superset consists, in the worst case, of the union of a circle and the mirror curve of the profile curve.