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

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.Comment: in Frenc

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

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.Comment: 33 pages, 17 figures, in French. typos corrected, references added. arXiv admin note: substantial text overlap with arXiv:1906.1001