A path in history, from curvature to convexity

Annette A'Campo-Neuen (1) , Athanase Papadopoulos (2)

1 Unibas - Université de Bâle = University of Basel = Basel Universität
2 IRMA - Institut de Recherche Mathématique Avancée

We describe a path in the history of curvature, starting from Greek antiquity, in the works of Euclid, Apollonius, Archimedes and a few others, passing through the works of Huygens, Euler, and Monge and his students, and ending in the twentieth century at the works of Bonnesen, Fenchel, Busemann, Feller and Alexandrov. Our goal is not to review the whole history of curvature, but to show how the approaches to curves, surfaces and curvature evolved from the synthetic point of view of the Greeks to the methods of analytic geometry founded by Fermat, Descartes, Newton and Leibniz, and eventually, in the twentieth century, experienced a return to the synthetic methods of the Greeks.

Mots clés: Curves; curvature; conics; Leonhard Euler; Christiaan Huygens; evolute; Busemann curvature; Alexandrov curvature; Dupin indicatrix; Dupin ellipsoid; osculating circle; Archimedes

Domaines: Histoire et perspectives sur les mathématiques [math.HO]; Topologie géométrique [math.GT]; Géométrie différentielle [math.DG]

Annette A'Campo-Neuen, Athanase Papadopoulos. A path in history, from curvature to convexity. In: Geometry in History (ed. S. G. Dani and A. Papadopoulos), Springer, p. 305-353, 2019, 978-3-030-13609-3. ⟨10.1007/978-3-030-13609-3⟩. ⟨hal-02441634⟩