Curves of Minimax Curvature

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.