On the total curvature of minimizing geodesics on convex surfaces
HTML articles powered by AMS MathViewer
- by N. Lebedeva and A. Petrunin
- St. Petersburg Math. J. 29 (2018), 139-153
- DOI: https://doi.org/10.1090/spmj/1485
- Published electronically: December 27, 2017
- PDF | Request permission
Abstract:
A universal upper bound is given for the total curvature of a minimizing geodesic on a convex surface in the Euclidean space.References
- Pankaj K. Agarwal, Sariel Har-Peled, Micha Sharir, and Kasturi R. Varadarajan, Approximating shortest paths on a convex polytope in three dimensions, J. ACM 44 (1997), no. 4, 567–584. MR 1481315, DOI 10.1145/263867.263869
- János Pach, Folding and turning along geodesics in a convex surface, Geombinatorics 7 (1997), no. 2, 61–65. MR 1487759
- Imre Bárány, Krystyna Kuperberg, and Tudor Zamfirescu, Total curvature and spiralling shortest paths, Discrete Comput. Geom. 30 (2003), no. 2, 167–176. U.S.-Hungarian Workshops on Discrete Geometry and Convexity (Budapest, 1999/Auburn, AL, 2000). MR 2007957, DOI 10.1007/s00454-003-0001-z
- J. Liberman, Geodesic lines on convex surfaces, C. R. (Doklady) Acad. Sci. URSS (N.S.) 32 (1941), 310–313. MR 0010994
- V. V. Usov, The length of the spherical image of a geodesic on a convex surface, Sibirsk. Mat. Ž. 17 (1976), no. 1, 233–236. (inside back cover) (Russian). MR 0405316
- I. D. Berg, An estimate on the total curvature of a geodesic in Euclidean $3$-space-with-boundary, Geom. Dedicata 13 (1982), no. 1, 1–6. MR 679213, DOI 10.1007/BF00149423
- A. V. Pogorelov, Vneshnyaya geometriya vypuklykh poverkhnosteĭ, Izdat. “Nauka”, Moscow, 1969 (Russian). MR 0244909
- V. A. Zalgaller, The question of the spherical representation of a geodesic, Ukrain. Geometr. Sb. 10 (1971), 12–18 (Russian). MR 0298604
- A. D. Milka, A shortest path with nonrectifiable spherical representation. I, Ukrain. Geometr. Sb. 16 (1974), 35–52, ii (Russian). MR 0385761
- V. V. Usov, The three-dimensional swerve of curves on convex surfaces, Sibirsk. Mat. Ž. 17 (1976), no. 6, 1427–1430, 1440 (Russian). MR 0442862
- Anton Petrunin, Applications of quasigeodesics and gradient curves, Comparison geometry (Berkeley, CA, 1993–94) Math. Sci. Res. Inst. Publ., vol. 30, Cambridge Univ. Press, Cambridge, 1997, pp. 203–219. MR 1452875
Bibliographic Information
- N. Lebedeva
- Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskii pr. 28, 198504 St. Petersburg, Russia; St. Petersburg Branch, Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia
- Email: lebed@pdmi.ras.ru
- A. Petrunin
- Affiliation: Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
- MR Author ID: 335143
- ORCID: 0000-0003-3053-5172
- Email: petrunin@math.psu.edu
- Received by editor(s): May 20, 2016
- Published electronically: December 27, 2017
- Additional Notes: N. Lebedeva was partially supported by RFBR grant 14-01-00062. A. Petrunin was partially supported by NSF grant DMS 1309340
- © Copyright 2017 American Mathematical Society
- Journal: St. Petersburg Math. J. 29 (2018), 139-153
- MSC (2010): Primary 53C21
- DOI: https://doi.org/10.1090/spmj/1485
- MathSciNet review: 3660688
Dedicated: Dedicated to Yu. D. Burago on the occasion of his 80th birthday