Iterating skew evolutes and skew involutes: a linear analog of the bicycle kinematics

TL;DR

This paper explores the geometry and dynamics of skew evolutes and involutes, which are linear analogs of bicycle kinematics, revealing their properties and relationships through mathematical analysis.

Contribution

It introduces the concepts of skew evolutes and involutes, analyzing their geometric and dynamic properties as a novel linear analog of bicycle kinematics.

Findings

Characterization of skew evolute and involute maps

Connection to bicycle kinematics and front-rear track analogy

Insights into the geometry and dynamics of these curves

Abstract

The evolute of a plane curve is the envelope of its normals. Replacing the normals by the lines that make a fixed angle with the curve yields a new curve, called the evolutoid. We prefer the term ``skew evolute", and we study the geometry and dynamics of the skew evolute map and of its inverse, the skew involute map. The relation between a curve and its skew evolute is analogous to the relation between the rear and front bicycle tracks, and this connections with the bicycle kinematics (a considerably more complicated subject) is our motivation for this study.