Circular evolutes and involutes of framed curves in the Euclidean space

TL;DR

This paper introduces the concepts of circular evolutes and involutes for framed curves in Euclidean space, exploring their properties, relationships, and singularities, extending classical notions to more general framed curves.

Contribution

It defines and analyzes circular evolutes and involutes for framed curves, generalizing classical evolutes and involutes, and studies their properties and singularities in Euclidean space.

Findings

Circular evolutes relate to curvature circles and singular value sets.

Involutes of framed curves generalize classical involutes of space curves.

Circular evolutes and involutes are inverse operations under certain conditions.

Abstract

We introduce circular evolutes and involutes of framed curves in the Euclidean space. Circular evolutes of framed curves stem from the curvature circles of Bishop directions and singular value sets of normal surfaces of Bishop directions. On the other hand, involutes of framed curves are direct generalizations of involutes of regular space curves and frontals in the Euclidean plane. We investigate properties of normal surfaces, circular evolutes, and involutes of framed curves. We can observe that taking circular evolutes and involutes of framed curves are opposite operations under suitable assumptions, similarly to evolutes and involutes of fronts in the Euclidean plane. Furthermore, we investigate the relations among singularities of normal surfaces, circular evolutes, and involutes of framed curves.