Offsets of a regular trifolium

TL;DR

This paper investigates the algebraic properties of offsets of a regular trifolium, focusing on different parametrizations and their impact on envelope computation, resulting in new symbolic and computational tools.

Contribution

It introduces methods for implicitizing offsets of a trifolium, demonstrating their algebraic degree and developing GeoGebra tools for automatic visualization.

Findings

Offset curve degree is 14 in the general case.

Implicit equations can be derived via different parametrizations.

GeoGebra applet visualizes offset curves automatically.

Abstract

The non-uniqueness of a rational parametrization of a rational plane curve may influence the process of computing envelopes of 1-parameter families of plane curves. We study envelopes of family of circles centred on a regular trifolium and its offsets, paying attention to different parametrizations. We use implicitization both to show that two rational parametrizations of a curve are equivalent, and to determine an implicit equation for the envelope under study. The derivation of an implicit equation of an offset follows another path, leading to new developments of the package GeoGebra Discovery. As an immediate symbolic result, we obtain that in the general case the offset curve of a regular trifolium is an algebraic curve of degree 14. We illustrate this fact by providing a GeoGebra applet that computes such curves automatically and visualizes them in a web browser.