A new method to compute the singularities of offsets to rational plane curves

TL;DR

This paper introduces a novel algebraic method to efficiently identify all real singularities of offset curves derived from rational plane curves, avoiding implicit equation computation.

Contribution

It provides a new approach to compute real singularities of offsets directly from the parametrization, enhancing efficiency and avoiding implicit equations.

Findings

Algorithm successfully computes all real singularities for moderate degrees.

Method avoids implicit equation computation, simplifying the process.

Experiments demonstrate the algorithm's efficiency in Maple.

Abstract

Given a planar curve defined by means of a real rational parametrization, we prove that the affine values of the parameter generating the real singularities of the offset are real roots of a univariate polynomial that can be derived from the parametrization of the original curve, without computing or making use of the implicit equation of the offset. By using this result, a finite set containing all the real singularities of the offset, and in particular all the real self-intersections of the offset, can be computed. We also report on experiments carried out in the computer algebra system Maple, showing the efficiency of the algorithm for moderate degrees.