Existence et unicit\'e d'une courbe \`a courbure positive maximisant le minimum du rayon de courbure

TL;DR

This paper proves the existence and uniqueness of a specific curve with positive curvature that maximizes the minimum radius of curvature, characterized as a combination of a circular arc and a line segment.

Contribution

It establishes the unique optimal curve with positive curvature maximizing the minimum radius of curvature, linking it to Dubins's curves and potential patent applications.

Findings

Unique maximizer curve is composed of a circular arc and a line segment.

The maximizer curve is the only one with these properties in set E.

Connection to Dubins's curves enhances understanding of optimal path design.

Abstract

We consider the set E of curves with positive algebraic curvature, whose extremities and tangents in their extremities are given. For each of the curves of E, we define the minimum of the radius of curvature. There exists a unique curve of E which maximizes this minimum and this curve is equal to the unique curve of E composed of an arc of circle and a line segment, where appropriate reduced to a point. This curve corresponds also to a particular case of Dubins's curve and will be used to improve the conception of a piece of a patent.