Abstract
Plane curves are studied both via their parametric equations or their Cartesian equation. We study the tangent to a curve and the related problem of the envelope of a family of curves; we exhibit some interesting applications in physics. After a careful study of the curvature of a plane curve, its intrinsic equation and the famous Umlaufsatz, we switch to the more involved question of simple closed and convex curves and we prove in particular the Hopf and the “Four vertices” theorems.
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References and Further Reading
F. Borceux, An Algebraic Approach to Geometry. Geometric Trilogy, vol. II (Springer, Berlin, 2014)
S.S. Chern, Differential Geometry. Lect. Notes (Dpt. Math. Univ., Chicago, 1954)
H. Hopf, Über die Drehung der Tangenten und Sehnen Ebener Kurven. Compos. Math. 2, 50–62 (1935)
J. Steiner, Einfache Beweise der Isoperimetrischen Hauptsätze. J. Reine Angew. Math. 18, 289–296 (1838)
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Borceux, F. (2014). Plane Curves. In: A Differential Approach to Geometry. Springer, Cham. https://doi.org/10.1007/978-3-319-01736-5_2
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DOI: https://doi.org/10.1007/978-3-319-01736-5_2
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-01735-8
Online ISBN: 978-3-319-01736-5
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