Focal surfaces and evolutes of framed curves in hyperbolic 3-space from the viewpoint of Legendrian duality

TL;DR

This paper explores the differential geometry of hyperbolic framed curves, focusing on focal surfaces, evolutes, and their singularities using Legendrian duality, revealing new relationships and classifications in hyperbolic 3-space.

Contribution

It introduces a novel approach to study focal surfaces and evolutes of hyperbolic framed curves via Legendrian duality, including singularity classification and duality relations.

Findings

Classification of singularities of dual surfaces

Relationship among focal surfaces, evolutes, and dual surfaces

Duality relations of singularities between focal surfaces and evolutes

Abstract

A hyperbolic framed curve is a smooth curve with a moving frame in hyperbolic 3-space. It may have singularities. By using this moving frame, we can investigate the differential geometry properties of curves, even at singular points. In fact, we consider the focal surfaces and evolutes of hyperbolic framed curves by using Legendrian dualities which developed by Chen and Izumiya. The focal surfaces are the dual surfaces of tangent indicatrix of original curves. Moreover, classifications of singularities of the serval dual surfaces are shown. By this, we give the relationship among focal surfaces, evolutes and dual surfaces of evolutes. Finally, we study duality relations of singularities between focal surfaces and dual surfaces of evolutes.