Generation of real algebraic loci via complex detours

TL;DR

This paper analyzes a locus generation algorithm in dynamic geometry software that uses complex detours to handle singularities, proposing a variant that can generate entire real algebraic loci and applying it to mechanical linkage simulation.

Contribution

It introduces a variant of the locus generation algorithm that potentially produces complete real algebraic loci and demonstrates its application in mechanical linkage simulation.

Findings

Algorithm is independent of complex detour orientation

Proposed variant can generate entire real algebraic loci

Application to mechanical linkages shows realistic motion

Abstract

We discuss the locus generation algorithm used by the dynamic geometry software Cinderella, and how it uses complex detours to resolve singularities. We show that the algorithm is independent of the orientation of its complex detours. We conjecture that the algorithm terminates if it takes small enough complex detours and small enough steps on every complex detour. Moreover, we introduce a variant of the algorithm that possibly generates entire real connected components of real algebraic loci. Several examples illustrate its use for organic generation of real algebraic loci. Another example shows how we can apply the algorithm to simulate mechanical linkages. Apparently, the use of complex detours produces physically reasonable motion of such linkages.