Pure and Applied Mathematics Quarterly

Volume 16 (2020)

Number 5

Symmetrization of convex plane curves

Pages: 1767 – 1787

DOI: https://dx.doi.org/10.4310/PAMQ.2020.v16.n5.a15

Authors

Peter Giblin (Department of Mathematical Sciences, University of Liverpool, United Kingdom)

Stanisław Janeczko (Institute of Mathematics, Polish Academy of Sciences, Warszawa, Poland; and Faculty of Mathematics and Information Science, Warsaw University of Technology, Warszawa, Poland)

Abstract

Several point symmetrizations of a convex curve $\Gamma$ are introduced and one, the affinely invariant ‘central symmetric transform’ (CST) with respect to a given basepoint inside $\Gamma$, is investigated in detail. Examples for $\Gamma$ include triangles, rounded triangles, ellipses, curves defined by support functions and piecewise smooth curves. Of particular interest is the region of basepoints for which the CST is convex (this region can be empty but its complement in the interior of $\Gamma$ is never empty). The (local) boundary of this region can have cusps and in principle it can be determined from a geometrical construction for the tangent direction to the CST.

Keywords

affine invariants, convex plane curves, singularities, symmetrization, higher inflections

2010 Mathematics Subject Classification

Primary 52A10. Secondary 53A04.

Full Text (PDF format)

Received 1 May 2020

Published 17 February 2021