On evolutoids of regular surfaces in Euclidean 3-space

TL;DR

This paper introduces the concept of evolutoids for regular surfaces in Euclidean 3-space, providing explicit parametrizations and analyzing their local behavior using singularity theory, thereby extending the planar curve concept to surfaces.

Contribution

It defines evolutoids for surfaces, offers explicit parametrizations, and studies their local properties through singularity theory, a novel extension from planar curves.

Findings

Explicit parametrization of surface evolutoids

Analysis of local behavior via singularity theory

Relations between surface geometry and evolutoids

Abstract

Inspired by the concept of evolutoids of planar curves, we present the concept of evolutoids for regular surfaces as an envelope of a two-parameter family of lines in Euclidean 3-space. We give an explicit parametrization for such evolutoids. Besides, we used the theory of singularities to study the local behavior of regular points of this object and presented some relations between the geometry of the regular surface and its evolutoid.