×
Coulon, Rémi; Matsumoto, Elisabetta A.; Segerman, Henry; Trettel, Steve J.

Ray-marching Thurston geometries. (English) Zbl 1517.81037

Exp. Math. 31, No. 4, 1197-1277 (2022).
Summary: We describe algorithms that produce accurate real-time interactive in-space views of the eight Thurston geometries using ray-marching. We give a theoretical framework for our algorithms, independent of the geometry involved. In addition to scenes within a geometry \(X\), we also consider scenes within quotient manifolds and orbifolds \(X / \Gamma\). We adapt the Phong lighting model to non-Euclidean geometries. The most difficult part of this is the calculation of light intensity, which relates to the area density of geodesic spheres. We also give extensive practical details for each geometry.

MSC:

81P68 Quantum computation
81V80 Quantum optics
68Q12 Quantum algorithms and complexity in the theory of computing
53Z05 Applications of differential geometry to physics
54B15 Quotient spaces, decompositions in general topology
70M20 Orbital mechanics
53A35 Non-Euclidean differential geometry
37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)

Software:

DLMF; Curved Spaces; OpenRelativity
Cite Review PDF
Full Text: DOI arXiv HAL

References:

[1] Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions, With Formulas, Graphs and Mathematical Tables (1966), U.S. Government Printing Office: U.S. Government Printing Office, Washington, DC
[2] Bradley, T. W.; Bradley, C. J.; Cracknell, A. P., The Mathematical Theory of Symmetry in Solids: Representation Theory for Point Groups and Space Groups (1972), Oxford: Clarendon Press, Oxford · Zbl 1180.20001
[3] Berger, P.
[4] Bridson, M. R.; Haefliger, A., Grundlehren der Mathematischen Wissenschaften, 319, Metric spaces of non-positive curvature (1999), Berlin: Springer-Verlag, Berlin · Zbl 0988.53001
[5] Berger, P.; Laier, A.; Velho, L., An image-space algorithm for immersive views in 3-manifolds and orbifolds, Vis. Comput, 31, 1, 93-104 (2015)
[6] Bonahon, F., Low-Dimensional Geometry, 49 (2009), Providence, RI: American Mathematical Society/ Princeton, Providence, RI · Zbl 1176.57001
[7] Bölcskei, A.; Szilágyi, B., Frenet formulas and geodesics in Sol geometry, Beiträge Algebra Geom, 48, 2, 411-421 (2007) · Zbl 1167.53021
[8] Bulirsch, R., Numerical calculation of elliptic integrals and elliptic functions, Numer. Math., 7, 78-90 (1965) · Zbl 0133.08702
[9] Culler, M.; Dunfield, N. M.; Goerner, M.; Weeks, J. R.
[10] Cooper, D.; Hodgson, C. D.; Kerckhoff, S. P., Chapter 2. Orbifolds, 5 (2000), Tokyo: The Mathematical Society of Japan, Tokyo · Zbl 0955.57014
[11] Coulon, R.; Matsumoto, E.; Segerman, H.; Trettel, S.; Yackel, Carolyn; Bosch, Robert; Torrence, Eve; Fenyvesi, Kristóf, Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture (Phoenix, Arizona), Non-euclidean virtual reality III: Nil, 153-160 (2020), Phoenix, Arizona: Tessellations Publishing, Phoenix, Arizona
[12] Coulon, R.; Matsumoto, E.; Segerman, H.; Trettel, S., Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture (Phoenix, Arizona), Non-euclidean virtual reality IV: Sol, 161-168 (2020), Tessellations Publishing
[13] Coulon, R.; Matsumoto, E. A.; Segerman, H.; Trettel, S.
[14] Coiculescu, M. P., Schwartz, R. E. (2019). The spheres of Sol, arXiv:1911.04003.
[15] Calvaruso, G.; Zaeim, A., Four-dimensional homogeneous lorentzian manifolds, Monatsh. für Math, 174 (2014) · Zbl 1301.53068
[16] Divjak, B.; Erjavec, Z.; Szabolcs, B.; Szilágyi, B., Geodesics and geodesic spheres in \(####\) geometry, Math. Commun., 14, 2, 413-424 (2009) · Zbl 1193.53103
[17] Egan, G., Dichronauts (2017), San Francisco: Night Shade Books, San Francisco
[18] Floyd, W.; Weber, B.; Weeks, J., The Achilles’ heel of \(####\)?, Experiment. Math, 11, 1, 91-97 (2002) · Zbl 1029.65017
[19] Geng, A. L. L. (2016), 5-Dimensional Geometries I: The general classification, arXiv:1605.07545.
[20] Goldman, W.
[21] Gregorcic, B.; Planinsic, G.; Etkina, E., Doing science by waving hands: Talk, symbiotic gesture, and interaction with digital content as resources in student inquiry, Phys. Rev. Phys. Educ. Res, 13, 2, 020104 (2017)
[22] Grayson, M. A., Geometry and growth in three dimensions (1983), ProQuest LLC. Ann Arbor: MI, Princeton University, ProQuest LLC. Ann Arbor
[23] Hart, V.; Hawksley, A.; Matsumoto, E.; Segerman, H., Non-euclidean virtual reality I: Explorations of ℍ^3, 33-40 (2017), Tessellations Publishing
[24] Hart, V.; Hawksley, A.; Matsumoto, E.; Segerman, H., Non-euclidean virtual reality II: Explorations of ℍ^2 × E, 41-48 (2017), Tessellations Publishing
[25] Hillman, J. A., Four-Manifolds, Geometries and Knots, 5 (2002), Geometry & Topology Publications · Zbl 1087.57015
[26] Hart, J. C.; Sandin, D. J.; Kauffman, L. H., Ray tracing deterministic 3-d fractals, SIGGRAPH Comput. Graph, 23, 3, 289-296 (1989)
[27] Jacobi, C. G. J., Fundamenta nova theoriae functionum ellipticarum (1829), Regiomonti
[28] Johnson-Glenberg, M. C.; Megowan-Romanowicz, C., Embodied science and mixed reality: How gesture and motion capture affect physics education, Cogn. Res.: Princ. Implic., 2, 1, 24 (2017)
[29] Kopczyński, E., Celińska-Kopczyńska, D. (2020). Real-time visualization in non-isotropic geometries, arXiv:2002.09533.
[30] Kopczyński, E.; Celińska, D.; Čtrnáct, M., HyperRogue: Playing with hyperbolic geometry, 9-16 (2017), Tessellations Publishing
[31] Kim, Y-b.; Oh, C. Y.; Park, N., Classical Geometry of de Sitter Space-Time: An Introductory Review, Journal of the Korean Physical Society, 42, 5, 573-592 (2003)
[32] Lawden, D. F., Elliptic Functions and Applications, 80 (1989), New York: Springer-Verlag. MR 1007595, New York · Zbl 0689.33001
[33] Lindgren, R.; Tscholl, M.; Wang, S.; Johnson, E., Enhancing learning and engagement through embodied interaction within a mixed reality simulation, Comp. Educ, 95, 174-187 (2016) · doi:10.1016/j.compedu.2016.01.001
[34] Luminet, J-P. (2019). An illustrated history of black hole imaging : Personal recollections (1972-2002), arXiv:1902.11196.
[35] MagmaMcFry, SolvView.
[36] Müller, T.; Grottel, S.; Weiskopf, D., Special relativistic visualization by local ray tracing, IEEE Trans. Visualiz. Comp. Graph, 16, 6, 1243-1250 (2010)
[37] Milnor, J., Curvatures of left invariant metrics on Lie groups, Adv. Math, 21, 3, 293-329 (1976) · Zbl 0341.53030
[38] Munzner, T.; Levy, S.; Phillips, M.; Fowler, C.; Gunn, C.; Thurston, N.; Krech, D.; Wisdom, S.; Meyer, D.; Rowley, T., Geomview: an interactive 3D viewing program for Unix (1991)
[39] Molnár, E., The projective interpretation of the eight 3-dimensional homogeneous geometries, Beiträge Algebra Geom, 38, 2, 261-288 (1997) · Zbl 0889.51021
[40] Molnár, E., On Nil geometry, Period. Polytech. Mech. Eng, 47, 1, 41-49 (2003) · Zbl 1084.53044
[41] McGrath, D.; Wegener, M.; McIntyre, T. J.; Savage, C.; Williamson, M., Student experiences of virtual reality: A case study in learning special relativity, Amer. J. Phys, 78, 862-868 (2010) · doi:10.1119/1.3431565
[42] Novello, T., da Silva, V., Velho, L. (2020a). Design and visualization of riemannian metrics, arXiv:2005.05386.
[43] Novello, T.; da Silva, V.; Velho, L., Global illumination of non-euclidean spaces, Comput. Graph, 93, 61-70 (2020) · doi:10.1016/j.cag.2020.09.014
[44] Novello, T.; da Silva, V.; Velho, L., Visualization of Nil, Sol, and \(####\) geometries, Comput. Graph., 91, 219-231 (2020)
[45] Nelson, R.; Segerman, H., Visualizing hyperbolic honeycombs, Journal of Mathematics and the Arts, 11, 1, 4-39 (2017) · Zbl 1405.52022 · doi:10.1080/17513472.2016.1263789
[46] Nelson, R.; Segerman, H.; Woodard, M., hypVR-Ray (2018)
[47] Olver, F. W. J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions (2010), Washington, DC: U.S. Department of Commerce, National Institute of Standards and Technology/ Cambridge: Cambridge University Press, Washington, DC: U.S. Department of Commerce, National Institute of Standards and Technology/ Cambridge · Zbl 1198.00002
[48] Oberhettinger, F.; Magnus, W., Anwendung der elliptischen Funktionen in Physik und Technik (1949), Berlin: Springer-Verlag, Berlin · Zbl 0034.33601
[49] Patrangenaru, V., Classifying 3- and 4-dimensional homogeneous riemannian manifolds by cartan triples, Pacific J. Math, 173, 2, 511-532 (1996) · Zbl 0866.53035
[50] Perelman, G. (2002). The entropy formula for the Ricci flow and its geometric applications, arXiv:0211159. · Zbl 1130.53001
[51] Perelman, G. (2003a). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:0307245. · Zbl 1130.53003
[52] Perelman, G. (2003b). Ricci flow with surgery on three-manifolds, arXiv:0303109. · Zbl 1130.53002
[53] Phillips, M.; Gunn, C., Visualizing hyperbolic space: Unusual uses of 4x4 matrices, 209-214 (1992), New York: Association for Computing Machinery, New York
[54] Phong, B. T., Illumination for computer generated pictures, Commun. ACM, 18, 6, 311-317 (1975) · doi:10.1145/360825.360839
[55] Quilez, I.
[56] Quilez, I.
[57] Scott, P., The geometries of 3-manifolds, Bullet. London Math. Soc, 15, 5, 401-487 (1983) · Zbl 0561.57001 · doi:10.1112/blms/15.5.401
[58] Sherin, Z. W.; Cheu, R.; Tan, P.; Kortemeyer, G., Visualizing relativity: The openrelativity project, Amer. J. Phys, 84, 369-374 (2016) · doi:10.1119/1.4938057
[59] Sokolowski, L. M., The bizarre anti-de Sitter spacetime (2016) · Zbl 1352.83007
[60] Savage, C. M.; Searle, A.; McCalman, L., Real time relativity: exploratory learning of special relativity, Amer. J. Phys, 75, 791-798 (2007) · doi:10.1119/1.2744048
[61] Thurston, W. P.; Levy, S., Three-Dimensional Geometry and Topology, 1 (1997), Princeton, NJ: Princeton University Press, Princeton, NJ · Zbl 0873.57001
[62] Thurston, W. P., How to see 3-Manifolds, Topology of the Universe Conference, Cleveland, 15, 2545-2571 (1998) · Zbl 0932.57017
[63] Trettel, S.
[64] Troyanov, M., L’horizon de Sol, Expos. Math, 16, 5, 441-479 (1998) · Zbl 0939.53042
[65] Velho, L.; da Silva, V.; Novello, T., Immersive visualization of the classical non-euclidean spaces using real-time ray tracing in VR, 423-430 (2020)
[66] Weeks, J.
[67] Weeks, J. (2020). Virtual reality simulations of curved spaces, arXiv:2011.00510.
[68] Weeks, J., Real-time rendering in curved spaces, IEEE Comp. Graph. Appl, 22, 6, 90-99 (2002)
[69] Wong, J.
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.