Moscow Mathematical Journal

Volume 23, Issue 4, October–December 2023 pp. 625–640.

Iterating Skew Evolutes and Skew Involutes: a Linear Analog of Bicycle Kinematics

Authors: Serge Tabachnikov (1)
Author institution:(1) Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA

Summary:

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 its skew evolute. We study the geometry and dynamics of the skew evolute maps and of their inverses, the skew involute maps. Among the motivations for this study are relations of this subject with tire track geometry and with mathematical billiards. We prove a version of the 4-vertex theorem where the role of circles is played by logarithmic spirals.

2020 Math. Subj. Class. 53A04, 53E99.

Keywords: Evolute, involute, evolutoid, bicycle kinematics, 4-vertex theorem.

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