Horocyclic evolutes, parallels and involutes of spacelike frontals in hyperbolic 2-space

TL;DR

This paper introduces new concepts of horocyclic parallels and involutes for spacelike frontals in hyperbolic 2-space, expanding the geometric framework using the enveloid theorem.

Contribution

It defines horocyclic parallels and involutes as normal envelopes of horocycles, establishing their relations with horocyclic evolutes in hyperbolic geometry.

Findings

Defined horocyclic parallels and involutes using the enveloid theorem

Established relations among horocyclic evolutes, parallels, and involutes

Extended the geometric theory of spacelike frontals in hyperbolic space

Abstract

The horocyclic evolutes of spacelike frontals in hyperbolic 2-space have already been defined. Using enveloid theorem, we now define the horocyclic parallel and involute of a spacelike frontal in hyperbolic 2-space as the normal envelopes of its normal and tangent horocycles, respectively. Meanwhile, we investigate the relations among horocyclic evolutes, parallels and involutes.