Barbier’s theorem in the Lobachevski plane
Author:
Jay P. Fillmore
Journal:
Proc. Amer. Math. Soc. 24 (1970), 705-709
MSC:
Primary 52.25; Secondary 53.00
DOI:
https://doi.org/10.1090/S0002-9939-1970-0253150-8
MathSciNet review:
0253150
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Abstract | References | Similar Articles | Additional Information
Abstract: In the Lobachevski plane, horocycles with the same center are geodesic parallels and are natural replacements for the lines used in defining the support function of a convex curve and the notion of constant width in the Euclidean plane. In this paper, analogs based on horocycles are obtained for Christoffel’s formula, which expresses the radius of curvature of a convex curve in terms of its support function, and Barbier’s theorem, which relates the length and width of a convex curve of constant width.
- W. J. Firey, The determination of convex bodies from their mean radius of curvature functions, Mathematika 14 (1967), 1–13. MR 217699, DOI https://doi.org/10.1112/S0025579300007956
- Detlef Laugwitz, Differential and Riemannian geometry, Academic Press, New York-London, 1965. Translated by Fritz Steinhardt. MR 0172184
- L. A. Santaló, Note on convex curves on the hyperbolic plane, Bull. Amer. Math. Soc. 51 (1945), 405–412. MR 12456, DOI https://doi.org/10.1090/S0002-9904-1945-08366-9
- L. A. Santaló, Horocycles and convex sets in hyperbolic plane, Arch. Math. (Basel) 18 (1967), 529–533. MR 225276, DOI https://doi.org/10.1007/BF01899495 P. A. Širokov, A sketch of the fundamentals of Lobachevskian geometry, Noordhoff, Groningen, 1964. MR 28 #4419.
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Additional Information
Keywords:
Convex curves,
constant width,
support function
Article copyright:
© Copyright 1970
American Mathematical Society