# Existence d'une courbe \`a courbure positive maximisant le minimum du rayon de courbure -- "Observation num\'erique"

## TL;DR

This paper investigates curves with positive curvature that maximize the minimum radius of curvature, proving existence theoretically and observing numerically that the optimal curve is a combination of a circle arc and a line segment, related to Dubins's curves.

## Contribution

It establishes the existence of an optimal curve with maximum minimal radius of curvature and identifies its structure as a combination of a circle arc and a line segment.

## Key findings

- Existence of a curve maximizing the minimum radius of curvature.
- Numerical observation that the optimal curve is a circle arc plus a line segment.
- Connection to Dubins's curves and potential applications in patent 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. We first prove that there exists a curve of E which maximizes this minimum. Numerically, we observe then that 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.