Curves of Minimax Spirality

TL;DR

This paper investigates optimal planar curves with minimal maximum curvature derivative, revealing they are composed of Euler spirals, circular arcs, and lines, with numerical methods demonstrated for specific cases.

Contribution

It characterizes the structure of minimax spirality curves using optimal control theory, including cases with curvature constraints and endpoint conditions.

Findings

Optimal curves are concatenations of Euler spirals, circular arcs, and lines.

When curvature constraints are inactive, optimal curves are Euler spirals or simple arcs.

Numerical methods effectively compute these minimax spirality curves.

Abstract

We study the problem of finding curves of minimum pointwise-maximum arc-length derivative of curvature, here simply called curves of minimax spirality, among planar curves of fixed length with prescribed endpoints and tangents at the endpoints. We consider the case when simple bounds (constraints) are also imposed on the curvature along the curve. The curvature at the endpoints may or may not be specified. We prove via optimal control theory that the optimal curve is some concatenation of Euler spiral arcs, circular arcs, and straight line segments. When the curvature is not constrained (or when the curvature constraint does not become active), an optimal curve is only made up of a concatenation of Euler spiral arcs, unless the oriented endpoints lie in a line segment or a circular arc of the prescribed length, in which case the whole curve is either a straight line segment or a circular arc segment, respectively. We propose numerical methods and illustrate these methods and the results by means of three example problems of finding such curves.