The Singular Evolutoids Set and the Extended Evolutoids Front

TL;DR

This paper introduces the singular evolutoid set for smooth planar curves and derives an integral equality for evolutoids using advanced geometric theorems, enriching the understanding of curve singularities.

Contribution

It defines the singular evolutoid set and applies the Gauss-Bonnet theorem to derive new integral relations for evolutoids of planar curves.

Findings

Defined the singular evolutoid set for planar curves.

Derived an integral equality for evolutoids using Gauss-Bonnet theorem.

Extended the concept to smooth periodic curves with cusp singularities.

Abstract

In this paper we introduce the notion of the singular evolutoid set which is the set of all singular points of all evolutoids of a fixed smooth planar curve with at most cusp singularities. By the Gauss-Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary (Theorem 2.20 in [4]) applied to the extended front of evolutoids of a hedgehog we obtain an integral equality for smooth periodic curves.