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Pedal curves of conics: an automated exploration of some cubics, sextics, octics and more

Thierry Dana-Picard

TL;DR

Pedal curves of conics are studied with automated methods to bridge geometry and algebra. The parabola case yields a cubic pedal curve with a complete position-based singularity classification; the ellipse case with an external pedal point produces a degree-$10$ curve that splits into quartic and sextic factors, prompting a Limaçon-inspired investigation. When the ellipse is pedal-referenced from the origin, the resulting degree-$20$ curve decomposes into multiple squared components (two quartics and two sextics), revealing a richer locus structure with extraneous components. The study demonstrates a productive workflow combining GeoGebra Discovery and Maple to analyze high-degree plane curves and highlights implications for pedagogy and future software integration.

Abstract

Constructions and exploration of plane algebraic curves has received a new push with the development of automated methods, whose algorithms are continuously improved and implemented in various software packages. We use them to explore the pedal curves of conics. This provides a construction of interesting geometric loci, given at first by parametric representations. After implicitization, they appear as sextics, octics and other curves of higher degree. We explore their irreducibility and their singular points (crunodes, cusps, etc.).

Pedal curves of conics: an automated exploration of some cubics, sextics, octics and more

TL;DR

Pedal curves of conics are studied with automated methods to bridge geometry and algebra. The parabola case yields a cubic pedal curve with a complete position-based singularity classification; the ellipse case with an external pedal point produces a degree- curve that splits into quartic and sextic factors, prompting a Limaçon-inspired investigation. When the ellipse is pedal-referenced from the origin, the resulting degree- curve decomposes into multiple squared components (two quartics and two sextics), revealing a richer locus structure with extraneous components. The study demonstrates a productive workflow combining GeoGebra Discovery and Maple to analyze high-degree plane curves and highlights implications for pedagogy and future software integration.

Abstract

Constructions and exploration of plane algebraic curves has received a new push with the development of automated methods, whose algorithms are continuously improved and implemented in various software packages. We use them to explore the pedal curves of conics. This provides a construction of interesting geometric loci, given at first by parametric representations. After implicitization, they appear as sextics, octics and other curves of higher degree. We explore their irreducibility and their singular points (crunodes, cusps, etc.).

Paper Structure

This paper contains 9 sections, 1 theorem, 18 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Pedal curves of a parabola
  3. Exploration with GeoGebra Discovery
  4. Algebraic work with Maple
  5. Pedal curves of a canonical ellipse with respect to an external point
  6. The ellipse is given by a canonical equation
  7. Algebraic work with Maple
  8. Pedal curves of a canonical ellipse with respect to the origin
  9. Discussion

Key Result

Theorem 2.1

Let $\mathcal{C}$ be a parabola and $D$ a point in the plane. We denote by $mathcal{P}$ the pedal curve of $mathcal{C}$ with respect to $D$.

Figures (7)

  • Figure 1: Pedal curve of a canonical parabola with respect to a point - the 3 cases
  • Figure 2: Pedal curve of a canonical parabola w.r.t a point, plotted with Maple
  • Figure 3: Construction Protocol of the pedal curve
  • Figure 4: Pedal curve of a canonical ellipse with respect to an external point - total geometric construction
  • Figure 5: Checking whether we obtained a Limaçon or not
  • ...and 2 more figures

Theorems & Definitions (2)

  • Definition 1.1
  • Theorem 2.1