Pedal curves of hyperbolic frontals and their singularities

TL;DR

This paper studies pedal curves of spacelike frontals in hyperbolic space, analyzing their singularities based on dual curve germs and Legendrian curvature, with illustrative examples.

Contribution

It introduces the concept of pedal curves for hyperbolic frontals and characterizes their singularities depending on dual curve germs and pedal point location.

Findings

Singularities depend on hyperbolic Legendrian curvature and pedal point location.

Different behaviors for non-singular and singular dual curve germs.

Examples illustrating the theoretical results.

Abstract

This paper introduces pedal curves of spacelike frontals in the hyperbolic 2-space. We mainly investigate the singularities of these hyperbolic pedal curves of spacelike frontals for non-singular and singular dual curve germs. We then show that for non-singular dual curve germs, the singularities of a pedal curve are dependent on the singularities of the first hyperbolic Legendrian curvature germ and the location of the pedal point, while for singular dual curve germs, they depend upon the singularities of both hyperbolic Legendrian curvature germs and also the location of the pedal point. We provide several examples with figures.