Curves of minimax spirality

¹ Mathematics, UniSA STEM, University of South Australia, Mawson Lakes, S.A. 5095, Australia
² Department of Mathematics and Statistics, The University of Western Australia, Nedlands WA 6009, Australia
³ School of Mathematics, Statistics, Chemistry and Physics, Murdoch University, Murdoch WA 6150, Australia

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Received: 16 September 2024
Accepted: 2 December 2025

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.