Local Shape of Generalized Offsets to Algebraic Curves

TL;DR

This paper investigates how generalized offsetting affects the local shape of algebraic curves, comparing it to classical offsets using differential geometry and local shape concepts.

Contribution

It provides a detailed analysis of local shape changes in algebraic curves under generalized offsetting, extending previous results on classical offsets.

Findings

Identifies conditions for local shape changes during generalized offsetting

Shows differences between classical and generalized offset effects

Uses differential geometry to analyze local behavior

Abstract

In this paper we study the local behavior of an algebraic curve under a geometric construction which is a variation of the usual offsetting construction, namely the {\it generalized} offsetting process (\cite {SS99}). More precisely, here we discuss when and how this geometric construction may cause local changes in the shape of an algebraic curve, and we compare our results with those obtained for the case of classical offsets (\cite{JGS07}). For these purposes, we use well-known notions of Differential Geometry, and also the notion of {\it local shape} introduced in \cite{JGS07}.