Envelopes of Circles Centered on a Kiss Curve
Thierry Dana-Picard, Daniel Tsirkin
TL;DR
This work addresses envelopes of 1-parameter families of plane curves and their offsets, focusing on circles of radius $R$ centered on the kiss curve given by $x=\cos u$, $y=\sin^3 u$, and comparing the resulting envelopes with the corresponding offsets. The approach blends Dynamic Geometry System exploration with Computer Algebra System methods (including Gröbner bases and elimination) to obtain both parametric and implicit descriptions and to locate singularities such as cusps and crunodes. The main contributions are the explicit parametric representations of the envelope components, the offset formulas with explicit parametrizations, and a systematic singular-point analysis, demonstrated for the kiss-curve case with numerical results. This technology-rich workflow clarifies how topology changes between envelopes and offsets arise from curvature and normal geometry, with broader implications for geometric locus problems and automated geometry tools.
Abstract
Envelopes of parameterized families of plane curves is an important topic, both for the mathematics involved and for its applications. Nowadays, it is generally studied in a technology-rich environment, and automated methods are developed and implemented in software. The exploration involves a dialog between a Dynamic Geometry System (used mostly for interactive exploration and conjectures) and a Computer Algebra System (for algebraic manipulations). We study envelopes of families of circles centered on the so-called kiss curve and offsets of this curve, observing the differences between constructs. Both parametric presentations and implicit equations are used, switching from parametric to polynomial representation being based on packages for Gröbner bases and Elimination. Singular points, both cusps and points of self-intersection (crunodes), are analyzed.