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A Bound on the Number of Endpoints of the Cut Locus

Published online by Cambridge University Press:  01 February 2010

Robert Sinclair  and
Minoru Tanaka
Robert Sinclair
Affiliation:
Department of Mathematical Sciences, Faculty of Science, University of the Ryukyus, Nishihara City, Okinawa 903–0213, Japan, [email protected], http://homepage.mac.com/r_m_sinclair/
Minoru Tanaka
Affiliation:
Department of Mathematics, Tokai University, Hiratsuka City, Kanagawa 259–1292, Japan, [email protected]

Abstract

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We provide strong experimental evidence for an upper bound on the number of endpoints of the cut locus from a point on a 2-surface of revolution. This bound is equal to the minimal number of intervals of monotone non-increasing or non-decreasing Gaussian curvature along one meridian from one pole to the other.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 9 , 2006 , pp. 21 - 39
Copyright
Copyright © London Mathematical Society 2006

References

1. Alexander, Stephanie B. and Bishop, Richard L., ‘Spines and homology of thin Riemannian manifolds with boundary’, Adv. Math. 155 (2000) 23–48. 2000]CrossRefGoogle Scholar
2. Alexander, Stephanie B. and Bishop, Richard L. ‘Spines and topology of thin Riemannian manifolds’, Trans. Amer. Math. Soc. 355 (2003) 4933–4954.CrossRefGoogle Scholar
3. Besse, Arthur L., ‘Manifolds all of whose geodesics are closed’, Ergeb. Math. Grenz-geb. 93 (Springer, Berlin, 1978).Google Scholar
4. Bleecker, David B., ‘Cut loci of closed surfaces without conjugate points’, Colloq. Math. 44 (1981) 263–276.CrossRefGoogle Scholar
5. Bliss, Gilbert A., ‘The geodesic lines on the anchor ring’, Ann. of Math. (2) 4 (1902) 1–21.CrossRefGoogle Scholar
6. Elerath, Doug, ‘An improved Toponogov comparison theorem for nonnegatively curved manifolds’, J. Differential Geom. 15 (1980) 187–216.CrossRefGoogle Scholar
7. Fleischmann, K., ‘Die geodätischen Linien auf Rotationsfiächen’, Inaugural Dissertation, Seyffarth, Liegnitz, 1915, http://www-gdz.sub.uni-goettingen.de/cgi-bin/digbib.cgi?PPN317487884.Google Scholar
8. Gluck, Herman and Singer, David A., ‘Scattering of geodesic fields, II’, Ann. of Math. 110 (1979) 205–225.CrossRefGoogle Scholar
9. Gooch, Bruce and Gooch, Amy, Non-photorealistic rendering (A. K. Peters, Natick, MA, 2001).CrossRefGoogle Scholar
10. Gravesen, Jens, Markvorsen, Steen, Sinclair, Robert and Tanaka, Minoru, ‘The cut locus of atoms of revolution’, Asian J. Math. 9 (2005) 103–120.CrossRefGoogle Scholar
11. Itoh, Jin-Ichi and Sinclair, Robert, ‘Thaw: a tool for approximating cut loci on a triangulation of a surface’, Experiment. Math. 13 (2004) 309–325.CrossRefGoogle Scholar
12. Jacobi, Carl Gustav Jakob, ‘Vorlesungen über Dynamik’, Gesammelte Werke, C. G. J. Jacobi, 2nd edn, supplementary volume (ed. Clebsch, A. and Lottner, E., Reimer, Georg, Berlin, 1884).Google Scholar
13. Kimball, B.F., ‘Geodesics on a toroid’, Amer. J. Math. 52 (1930) 29–52.CrossRefGoogle Scholar
14. Ma, Chuan Yu, ‘The number of vertices of cut loci C(A) of closed Riemannian surfaces’, Nanjing Daxue Xuebao Shuxue Bannian Kan 4 (1987) 106–108.Google Scholar
15. Mongoldt, Hans Von, ‘Ueber die jenigen Punkte auf positiv gekrümmten Flächen, welche die Eigenschaft haben, dass die von ihnen ausgehenden geodätischen Liniennie aufhören, kürzeste Linien zu sein’, J. Reine Angew. Math. 91 (1881) 23–53.CrossRefGoogle Scholar
16. Margerin, Christophe M., ‘General conjugate loci are not closed’, Differential geometry: Riemannian geometry, ed. Greene, Robert and Yau, ST., Proc. Symposia in Pure Math. 54, Part 3 (Amer. Math. Soc, Providence, RI, 1993) 465–478.CrossRefGoogle Scholar
17. Myers, Sumner Byron, ‘Connections between differential geometry and topology: I.Simply connected surfaces’, Duke Math. J. 1 (1935) 376–391.CrossRefGoogle Scholar
18. Myers, Sumner Byron, ‘Connections between differential geometry and topology: II.Closed surfaces’, Duke Math. J. 2 (1936) 95–102.CrossRefGoogle Scholar
19. Poincaré, Henri, ‘Sur les Lignes géodésiques des surfaces convexes’, Trans. Amer. Math. Soc. 6 (1905) 237–274.Google Scholar
20. Rebel, Johannes, ‘Tower of Babylon’, MSc Thesis, Eksamensprojekt nr. 1995–07, Mathematical Institute, Technical University of Denmark, 1995.Google Scholar
21. Saito, Takafumi and Takahashi, Tokiichiro, ‘Comprehensible rendering of 3-D shapes’, Computer Graphics 24, Number 4 (1990) 197–206.CrossRefGoogle Scholar
22. Sakai, Takashi, Riemannian geometry, Transl. Math. Monogr. 149 (Amer. Math. Soc, Providence, RI, 1996).Google Scholar
23. Shiohama, Katsuhiro, Shioya, Takashi and Tanaka, Minoru, The geometry of total curvature on complete open surfaces, Cambridge Tracts in Math. 159 (Cambridge University Press, 2003).CrossRefGoogle Scholar
24. Sinclair, Robert and Tanaka, Minoru, ‘Loki: software for computing cut loci’, Experiment. Math. 11 (2002) 1–25.CrossRefGoogle Scholar
25. Sinclair, Robert, ‘On the last geometric statement of Jacobi’, Experiment. Math. 12 (2003) 477–485.CrossRefGoogle Scholar
26. Sinclair, Robert and Tanaka, Minoru, ‘The cut locus of a 2-sphere of revolution and Toponogov's comparison theorem’, preprint, http://homepage.mac.com/r_m_sinclair/Cutlocus2sphere_draft.pdf.Google Scholar
27. Struik, Dirk J., Lectures on classical differential geometry, 2nd edn (Dover Publications, New York, 1988).Google Scholar
28. Tamura, Kazuhiro, On the cut locus of a complete Riemannian manifold homeomorphic to a cylinder, MSc Thesis, Graduate Course of Science, Tokai University, Japan, 2004.Google Scholar
29. Tanaka, Minoru, ‘On the cut loci of a von Mangoldt's surface of revolution’, J. Math. Soc. Japan 44 (1992) 631–641.CrossRefGoogle Scholar
30. Tanaka, Minoru, ‘On a characterization of a surface of revolution with many poles’, Mem. Fac. Set, Kyushu Univ. Ser. A, Mathematics 46 (1992) 251–268.Google Scholar
31. Tsuji, Yuichi, ‘On a cut locus of a complete Riemannian manifold homeomorphic to cylinder’, Proc. School of Sci., Tokai Univ. 32 (1997) 23–34.Google Scholar
Supplementary material: File

Sinclair and Tanaka Appendix

Appendix

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