Pedals and inversions of quadratic curves

TL;DR

This paper studies the pedal and primitive of quadratic curves in the Euclidean plane, providing a characterization that generalizes limacons of Pascal, and explores their singular points and geometric properties.

Contribution

It introduces a new characterization of pedals of quadratic curves, extending classical results and connecting them to limacons of Pascal.

Findings

Characterization of pedals of quadratic curves

Generalization of limacons of Pascal

Analysis of singular points at inflections

Abstract

The pedal of a curve in the Euclidean plane is a classical subject which has a singular point at the inflection point of the original curve or the pedal point. The primitive of a curve is a curve given by the inverse construction for making the pedal. In this paper we consider the pedal of a quadratic curve. On of the main results gives a characterization of such curves, which is one of the generalizations of limacons of Pascal.