Exploration of offsets of Cayley ovals and their singularities

TL;DR

This paper investigates the geometric and topological properties of offsets of Cayley ovals, revealing intriguing features like cusps and self-intersections through computational methods and geometric analysis.

Contribution

It introduces a detailed study of Cayley oval offsets, highlighting their unique properties and the effects of varying offset distances, supported by automated computational techniques.

Findings

Offsets exhibit cusps and self-intersections.

Shape and topology change significantly with offset distance.

Automated software aids in geometric analysis.

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.