A geometric invariant for the study of planar curves and its application to spiral tip meander

TL;DR

This paper introduces a geometric invariant based on total curvature for analyzing planar curves, demonstrating its utility through examples and applying it to enhance models of spiral wave meander.

Contribution

It presents a novel method to compute total curvature of periodic planar curves without reparameterization, aiding in scientific modeling.

Findings

Total curvature constrains symmetry in periodic curves.

The method simplifies curvature computation for complex curves.

Application improves understanding of spiral wave dynamics.

Abstract

Planar curves with periodically varying curvature arise in the natural sciences as the result of a wide variety of periodic processes. The total curvature of a periodic arc in such curves constrains their symmetry. It is shown how the total curvature can be computed without reparameterizing the curve to unit speed. The use of the total curvature of the periodic arcs is demonstrated through a series of four examples from various branches of science. Insights gained from these examples are applied to improve the modeling of spiral wave meander.