Curves of Minimax Curvature

TL;DR

This paper addresses the problem of finding planar curves with fixed length and boundary conditions that minimize the maximum curvature, using optimal control theory and numerical methods to classify solutions.

Contribution

It reformulates the minimax curvature problem as an optimal control problem and develops a numerical approach to classify and compute solutions.

Findings

Classification of solution types based on geometrical and control-theoretic analysis

Reduction of the infinite-dimensional problem to a finite-dimensional one with six variables

Numerical examples illustrating the solution structures

Abstract

We consider the problem of finding curves of minimum pointwise-maximum curvature, i.e., curves of minimax curvature, among planar curves of fixed length with prescribed endpoints and tangents at the endpoints. We reformulate the problem in terms of optimal control and use the maximum principle, as well as some geometrical arguments, to produce a classification of the types of solutions. Using the classification, we devise a numerical method which reduces the infinite-dimensional optimization problem to a finite-dimensional problem with just six variables. The solution types, together with some further observations on optimality, are illustrated via numerical examples.