Evolutes and involutes of framed curves in Euclidean 3-space

TL;DR

This paper defines and analyzes evolutes and involutes of framed curves in Euclidean 3-space, including conditions for their inverse relationship and cases where they are not inverses.

Contribution

It introduces a direct definition of evolutes and involutes for non-degenerate and framed curves using Bertrand type curve theory, expanding existing geometric concepts.

Findings

Conditions under which evolutes and involutes are inverse operations.

Analysis of cases where evolutes and involutes are not inverses.

Extension of evolute and involute theory to curves with singular points.

Abstract

We investigated the evolute of a space curve with singular points. As smooth curves with singular points, we apply the theory of framed curves. However, the involute corresponding to the evolute in the sense of the locus of the centre of osculating spheres has not been defined as far as we know. In this paper, we directly define the evolutes and involutes of non-degenerate curves and framed curves using the theory of Bertrand type curves. We give conditions that the evolutes and involutes are inverse operations of these curves. Moreover, we investigate the other cases in which the evolutes and involutes are not inverse operations.