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

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.).