Envelopes of Circles Centered on a Kiss Curve

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 Gr\"obner bases and Elimination. Singular points, both cusps and points of self-intersection (crunodes), are analyzed.