Exploration of offsets of Cayley ovals and their singularities
Thierry Dana-Picard, Daniel Tsirkin
TL;DR
This work investigates offsets of Cayley ovals, a non-singular plane curve family, by combining dynamic geometry software and computer algebra systems to analyze how offset distance $d$ and eccentricity $e$ affect shape and topology. The authors delineate envelopes versus offsets, explore the octic Cayley ovals across four $e$-regimes, and deploy derivative- and curvature-based methods as well as numerical solvers to locate cusps and crunodes. A key contribution is demonstrating that offset topology can differ markedly from the progenitor, including the emergence of multiple loops, crunodes, and complex singularities, even when the progenitor is non-singular. The study also documents practical workflow considerations for coordinating GeoGebra, LocusEquation in GD, and Maple for robust geometric analysis, with emphasis on the nontrivial interplay between algebraic and geometric representations and potential engineering applications of offsets.
Abstract
We explore offsets of Cayley ovals, by networking with different kinds of software. Using their specific abilities, algebraic, geometric, dynamic, we conjecture interesting properties of the offsets. For a given progenitor (the given plane curve whose offsets are studied), changes in the offset distance induce great changes in the shape and the topology of the offset. Such a study has been performed in the past for classical curves, and recently for non classical ones.Here we relate to Cayley ovals; despite them being non singular, their offsets have intriguing properties, cusps, and self-intersections. We begin with a short study of envelopes of families of circles with constant radius centered on the oval (these constructs are often studied together with offsets, but they are different objects). Then we study the offsets, which are defined as geometric loci. Both approaches are supported by the automated methods provided by the software.