Computing roadmaps in unbounded smooth real algebraic sets I: connectivity results

TL;DR

This paper extends the computation of roadmaps in real algebraic sets to unbounded cases by utilizing properties of generalized polar varieties, enhancing connectivity analysis in algebraic geometry applications like robotics.

Contribution

It introduces new connectivity results for unbounded smooth real algebraic sets using generalized polar varieties, removing the boundedness restriction in roadmap algorithms.

Findings

Extended connectivity properties to unbounded sets.

Utilized generalized polar varieties for critical locus analysis.

Improved roadmap computation methods for unbounded algebraic sets.

Abstract

Answering connectivity queries in real algebraic sets is a fundamental problem in effective real algebraic geometry that finds many applications in e.g. robotics where motion planning issues are topical. This computational problem is tackled through the computation of so-called roadmaps which are real algebraic subsets of the set V under study, of dimension at most one, and which have a connected intersection with all semi-algebraically connected components of V. Algorithms for computing roadmaps rely on statements establishing connectivity properties of some well-chosen subsets of V , assuming that V is bounded. In this paper, we extend such connectivity statements by dropping the boundedness assumption on V. This exploits properties of so-called generalized polar varieties, which are critical loci of V for some well-chosen polynomial maps.