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Loki: software for computing cut loci. (English) Zbl 1052.53001

The paper describe the software tool Loki for computing cut loci. It can display the cut locus from a point on a genus-1 two-dimensional Riemannian manifold. This manifold can be defined via a parametrization of an embedding in \({\mathbb R}^3\), or via an explicit metric. The code is freely available and it should be able to be extendable to compute conjugate loci. Loki has been created with ease of use in mind and the algorithm takes into account the global nature of the problem.

MSC:

53-04 Software, source code, etc. for problems pertaining to differential geometry
53C20 Global Riemannian geometry, including pinching

Software:

Loki
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References:

[1] Aurenhammer F., ACM Computing Surveys. 23 (3) pp 345– (1991)
[2] Barth T. J., J. Comp. Physics. 145 pp 1– (1998) · Zbl 0911.65091
[3] Berger M., Differential Geometry: Manifolds, Curves, and Surfaces (1988) · Zbl 0629.53001
[4] Berger M., University Lecture Series 17, in: Riemannian geometry during the second half of the twentieth century (2000)
[5] Bishop R. L., Proc. of the AMS. 65 pp 133– (1977)
[6] Bleecker D. D., Colloquium Mathematicum. 44 pp 263– (1981)
[7] Buchner M. A., Compositio Mathematica. 37 pp 103– (1978)
[8] Chavel I., Riemannian Geometry : A Modern Introduction (1993)
[9] do Carmo M. P., Differential Geometry of Curves and Surfaces (1976) · Zbl 0326.53001
[10] do Carmo M. P., Riemannian Geometry (1992)
[11] Degen W. L.F., Geometric Dedicata. 67 pp 197– (1997) · Zbl 0888.52008
[12] Gluck H., Ann. of Math., II. Ser. 108 pp 347– (1978) · Zbl 0399.58011
[13] Gluck H., Ann. of Math., II. Ser. 110 pp 205– (1979) · Zbl 0436.53047
[14] Hartman P., Amer. J. Math. 86 pp 705– (1964) · Zbl 0128.16105
[15] Hebda J. J., J. Diff. Geometry. 40 pp 621– (1994)
[16] Heck A., Introduction to Maple, (1996) · Zbl 0861.65001
[17] IEEE Standard for Binary Floating Point Arithmetic (1985)
[18] Itoh J., Tohoku Math. J. 50 pp 571– (1998) · Zbl 0939.53029
[19] Itoh J., Trans. Amer. Math. Soc. 353 (1) pp 21– (2001) · Zbl 0971.53031
[20] Kimmel R., Proc. of the Natl. Acad. Sciences USA. 95 pp 8431– (1998) · Zbl 0908.65049
[21] Kimmel, R. and Sethian, J. A. 1999. ”Fast Voronoi Diagrams and Offsets on Triangulated Surfaces.”. Proceedings of A FA Conference on Curves and Surfaces. July1999, Saint-Malo, France. [Kimmel and Sethian, 99]
[22] Klingenberg W., Riemannian Geometry (1982)
[23] Kunze, R., Wolter, F.E. and Rausch, T. ”Geodesic Voronoi Diagrams on Parametric Surfaces.”. Proceedings of CGI 1997, IEEE, Computer Society Press Conference Proceedings. pp.230–237. IEEE. [Kunze, et al. 97]
[24] Lawall J., Partial Evaluation: Practice and Theory pp 338– (1998)
[25] Maekawa T., J. of Mech. Design. 118 pp 499– (1996)
[26] Myers S. B., Duke Math. J. 1 pp 376– (1935) · Zbl 0012.27502
[27] Myers S. B., Duke Math. J. 2 pp 95– (1936) · Zbl 0013.32201
[28] Ozols V., Tohoku Mathematical Journal, II. Ser. 26 pp 219– (1974) · Zbl 0285.53034
[29] Rall L. B., Automatic Differentiation: Techniques and Applications (1981) · Zbl 0473.68025
[30] Rausch, T., Wolter, F.E. and Niehotta, O. Computation of Medial Curves on Surfaces. Proceedings of Conference on the Mathematics of Surfaces VII, IMA Conference Series. pp.43–68. [Rausch et al. 97] · Zbl 0965.65021
[31] Rebel J., Master of Science Thesis, in: Tower of Babylon (1995)
[32] Sakai T., Riemannian Geometry, Translation of Mathematical Monographs 149 (1992)
[33] Sethian J. A., Level Set Methods and Fast Marching Methods (1999) · Zbl 0929.65066
[34] Shiohama K., Collection SMF No.l, Actes de la table ronde de Géométrie différentielle en l’honneur Marcel Berger pp 531– (1996)
[35] Stroustrup B., The C++ Programming Language,, 3. ed. (1997) · Zbl 0825.68056
[36] Tanaka M., J. Math. Soc. Japan. 44 pp 631– (1992) · Zbl 0789.53023
[37] Taubin G., IEEE Computer Graphics and Applications. 14 pp 14– (1994) · Zbl 05085747
[38] Tsuji Y., Proceedings of the School of Science of Tokai University. 32 pp 23– (1997)
[39] Warren M. S., Supercomputing 1992 pp 570– (1992)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.