Dynamic constructions of hyperbolisms of plane curves: an automated exploration of geometric loci

TL;DR

This paper explores the construction of hyperbolisms of plane curves using automated software, deriving explicit equations for complex curves like quartics and sextics, and introduces a new method for constructing a lemniscate of Gerono.

Contribution

It presents a novel automated approach to construct and analyze hyperbolisms of plane curves, including explicit parametric and polynomial equations for complex algebraic curves.

Findings

Derived parametric equations for specific curves

Generated polynomial equations enabling irreducibility checks

Automated exploration of hyperbolisms for complex curves

Abstract

Hyperbolism of a given curve with respect to a point and a line is an interesting construct, a special kind of geometric locus, not frequent in the literature. While networking between two different kinds of mathematical software, we explore various cases, involving quartics, among them the so-called Kuelp quartic and topologically equivalent curves, and also an example with a sextic and a curve of degree 12. By a similar but different way, we derive a new construction of a lemniscate of Gerono. First, parametric equations are derived for the curve, then we perform implicitization Groebner bases packages and using elimination. The polynomial equation which is obtained enables to check irreducibility of the constructed curve.