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Special Issue on Illustration and Visualization in Mathematics Research

Ray-Marching Thurston Geometries

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References

  • Handbook of Mathematical Functions, With Formulas, Graphs and Mathematical Tables, ed. Abramowitz, M., Stegun, I. A. Fifth printing, with corrections. National Bureau of Standards Applied Mathematics Series, Vol. 55, National Bureau of Standards, Washington, DC (for sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, DC, 20402), 1966. MR 0208798
  • Bradley, T. W., Bradley, C. J., Cracknell, A. P. (1972). The Mathematical Theory of Symmetry in Solids: Representation Theory for Point Groups and Space Groups. Oxford: Clarendon Press.
  • Berger, P. Espaces Imaginaires. Available at: http://espaces-imaginaires.fr, 2015.
  • Bridson, M. R., Haefliger, A. Metric spaces of non-positive curvature. In: Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Vol. 319. Berlin: Springer-Verlag, 1999. MR 1744486
  • Berger, P., Laier, A., Velho, L. (2015). An image-space algorithm for immersive views in 3-manifolds and orbifolds. Vis. Comput. 31(1):93–104.
  • Bonahon, F. (2009). Low-Dimensional Geometry. Student Mathematical Library, Vol. 49. Providence, RI: American Mathematical Society/ Princeton, NJ: Institute for Advanced Study (IAS), From Euclidean surfaces to hyperbolic knots, IAS/Park City Mathematical Subseries. MR 2522946
  • Bölcskei, A., Szilágyi, B. (2007). Frenet formulas and geodesics in Sol geometry. Beiträge Algebra Geom. 48(2):411–421. MR 2364799
  • Bulirsch, R. (1965). Numerical calculation of elliptic integrals and elliptic functions. Numer. Math. 7:78–90. MR 175284
  • Culler, M., Dunfield, N. M., Goerner, M., Weeks, J. R. SnapPy, a computer program for studying the geometry and topology of 3-manifolds. Available at: http://snappy.computop.org (06/16/2020, version 2.8).
  • Cooper, D., Hodgson, C. D., Kerckhoff, S. P. (2000). Chapter 2. Orbifolds. MSJ Memoirs, Vol. 5, Tokyo: The Mathematical Society of Japan.
  • Coulon, R., Matsumoto, E., Segerman, H., Trettel, S. (2020a), Non-euclidean virtual reality III: Nil. In: Yackel, Carolyn and Bosch, Robert and Torrence, Eve and Fenyvesi, Kristóf Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture (Phoenix, Arizona), Phoenix, Arizona: Tessellations Publishing, pp. 153–160.
  • Coulon, R., Matsumoto, E., Segerman, H., Trettel, S. (2020b). Non-euclidean virtual reality IV: Sol, In: Proceedings of Bridges 2020: Mathematics, Art, Music, Architecture, Education, Culture (Phoenix, Arizona), Tessellations Publishing, pp. 161–168.
  • Coulon, R., Matsumoto, E. A., Segerman, H., Trettel, S. Non-euclidean VR. Available at: https://github.com/henryseg/non-euclidean_VR, 2020.
  • Coiculescu, M. P., Schwartz, R. E. (2019). The spheres of Sol, arXiv:1911.04003.
  • Calvaruso, G., Zaeim, A. (2014). Four-dimensional homogeneous lorentzian manifolds. Monatsh. für Math. 174,
  • Divjak, B., Erjavec, Z., Szabolcs, B., Szilágyi, B. (2009). Geodesics and geodesic spheres in SL(2,ℝ)˜ geometry. Math. Commun. 14(2): 413–424. MR 2743187
  • Egan, G. (2017). Dichronauts. San Francisco: Night Shade Books.
  • Floyd, W., Weber, B., Weeks, J. (2002). The Achilles’ heel of O(3,1)? Experiment. Math. 11(1): 91–97. MR 1960304
  • Geng, A. L. L. (2016), 5-Dimensional Geometries I: The general classification, arXiv:1605.07545.
  • Goldman, W. Geometric structures on manifolds. Available at: http://www.math.umd.edu/∼wmg/gstom.pdf.
  • Gregorcic, B., Planinsic, G., Etkina, E. (2017). 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.
  • Grayson, M. A. (1983). Geometry and growth in three dimensions, Thesis (Ph.D.), ProQuest LLC. Ann Arbor, MI, Princeton University. MR 2632777
  • Hart, V., Hawksley, A., Matsumoto, E., Segerman, H. (2017a). Non-euclidean virtual reality I: Explorations of ℍ3, Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture (Phoenix, Arizona), Tessellations Publishing. Available at: http://archive.bridgesmathart.org/2017/bridges2017-33.pdf, pp. 33–40.
  • Hart, V., Hawksley, A., Matsumoto, E., Segerman, H. (2017b). Non-euclidean virtual reality II: Explorations of ℍ2 × E, Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture (Phoenix, Arizona), Tessellations Publishing. Available at: http://archive.bridgesmathart.org/2017/bridges2017-41.pdf, pp. 41–48.
  • Hillman, J. A. (2002). Four-Manifolds, Geometries and Knots, Geometry & Topology Monographs, Vol. 5. Coventry: Geometry & Topology Publications. MR 1943724
  • Hart, J. C., Sandin, D. J., Kauffman, L. H. (1989). Ray tracing deterministic 3-d fractals. SIGGRAPH Comput. Graph. 23(3):289–296.
  • Jacobi, C. G. J. (1829). Fundamenta nova theoriae functionum ellipticarum, Regiomonti.
  • Johnson-Glenberg, M. C., Megowan-Romanowicz, C. (2017). Embodied science and mixed reality: How gesture and motion capture affect physics education. Cogn. Res.: Princ. Implic. 2(1):24.
  • Kopczyński, E., Celińska-Kopczyńska, D. (2020). Real-time visualization in non-isotropic geometries, arXiv:2002.09533.
  • Kopczyński, E., Celińska, D., Čtrnáct, M. (2017). HyperRogue: Playing with hyperbolic geometry, Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture (Phoenix, Arizona), Tessellations Publishing. Available at: http://archive.bridgesmathart.org/2017/bridges2017-9.pdf, pp. 9–16.
  • Kim, Y-b., Oh, C. Y., Park, N. (2003). Classical Geometry of de Sitter Space-Time: An Introductory Review. Journal of the Korean Physical Society, Vol. 42, No. 5, pp. 573–592.
  • Lawden, D. F. (1989). Elliptic Functions and Applications, Applied Mathematical Sciences, Vol. 80, New York: Springer-Verlag. MR 1007595
  • Lindgren, R., Tscholl, M., Wang, S., Johnson, E. (2016). Enhancing learning and engagement through embodied interaction within a mixed reality simulation. Comp. Educ. 95, 174–187. doi:10.1016/j.compedu.2016.01.001
  • Luminet, J-P. (2019). An illustrated history of black hole imaging : Personal recollections (1972–2002), arXiv:1902.11196.
  • MagmaMcFry, SolvView. Available at: https://github.com/MagmaMcFry/SolvView, 2019.
  • Müller, T., Grottel, S., Weiskopf, D. (2010). Special relativistic visualization by local ray tracing. IEEE Trans. Visualiz. Comp. Graph. 16(6):1243–1250.
  • Milnor, J. (1976). Curvatures of left invariant metrics on Lie groups. Adv. Math. 21(3):293–329. MR 425012
  • Munzner, T., Levy, S., Phillips, M., Fowler, C., Gunn, C., Thurston, N., Krech, D., Wisdom, S., Meyer, D., Rowley, T. (1991–2014). Geomview: an interactive 3D viewing program for Unix. Available at: http://www.geomview.org.
  • Molnár, E. (1997). The projective interpretation of the eight 3-dimensional homogeneous geometries. Beiträge Algebra Geom. 38(2):261–288. MR 1473106
  • Molnár, E. (2003). On Nil geometry. Period. Polytech. Mech. Eng. 47(1):41–49. MR 2045762
  • McGrath, D., Wegener, M., McIntyre, T. J., Savage, C., Williamson, M. (2010). Student experiences of virtual reality: A case study in learning special relativity. Amer. J. Phys. 78: 862–868. doi:10.1119/1.3431565
  • Novello, T., da Silva, V., Velho, L. (2020a). Design and visualization of riemannian metrics, arXiv:2005.05386.
  • Novello, T., da Silva, V., Velho, L. (2020b). Global illumination of non-euclidean spaces. Comput. Graph. 93:61–70. doi:10.1016/j.cag.2020.09.014
  • Novello, T., da Silva, V., Velho, L. (2020c), Visualization of Nil, Sol, and SL˜(2,ℝ) geometries, Comput. Graph. 91:219–231.
  • Nelson, R., Segerman, H. (2017). Visualizing hyperbolic honeycombs, Journal of Mathematics and the Arts 11(1):4–39. doi:10.1080/17513472.2016.1263789
  • Nelson, R., Segerman, H., Woodard, M. (2018). hypVR-Ray, Available at: https://github.com/mtwoodard/hypVR-Ray.
  • Olver, F. W. J., Lozier, D. W., Boisvert, R. F., Clark, C. W. (eds.) (2010). NIST Handbook of Mathematical Functions, Washington, DC: U.S. Department of Commerce, National Institute of Standards and Technology/ Cambridge: Cambridge University Press. With 1 CD-ROM (Windows, Macintosh and UNIX). MR 2723248
  • Oberhettinger, F., Magnus, W. (1949). Anwendung der elliptischen Funktionen in Physik und Technik. Berlin: Springer-Verlag. MR 0031129
  • Patrangenaru, V. (1996). Classifying 3- and 4-dimensional homogeneous riemannian manifolds by cartan triples. Pacific J. Math. 173(2):511–532.
  • Perelman, G. (2002). The entropy formula for the Ricci flow and its geometric applications, arXiv:0211159.
  • Perelman, G. (2003a). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:0307245.
  • Perelman, G. (2003b). Ricci flow with surgery on three-manifolds, arXiv:0303109.
  • Phillips, M., Gunn, C. (1992). Visualizing hyperbolic space: Unusual uses of 4x4 matrices, Proceedings of the 1992 Symposium on Interactive 3D Graphics (I3D ’92). New York: Association for Computing Machinery, pp. 209–214.
  • Phong, B. T. (1975). Illumination for computer generated pictures. Commun. ACM 18(6):311–317. doi:10.1145/360825.360839
  • Quilez, I. (xxxx). Distance functions. Available at: https://iquilezles.org/www/articles/distfunctions/distfunctions.htm.
  • Quilez, I. (xxxx). Soft shadows in raymarched SDFs. Available at: https://www.iquilezles.org/www/articles/rmshadows/rmshadows.htm.
  • Scott, P. (1983). The geometries of 3-manifolds. Bullet. London Math. Soc. 15(5):401–487 (English). doi:10.1112/blms/15.5.401
  • Sherin, Z. W., Cheu, R., Tan, P., Kortemeyer, G. (2016). Visualizing relativity: The openrelativity project. Amer. J. Phys. 84:369–374. doi:10.1119/1.4938057
  • Sokolowski, L. M. (2016). The bizarre anti-de Sitter spacetime. Available at: https://arxiv.org/pdf/1611.01118.pdf.
  • Savage, C. M., Searle, A., and McCalman, L. (2007). Real time relativity: exploratory learning of special relativity. Amer. J. Phys. 75:791–798. doi:10.1119/1.2744048
  • Thurston, W. P. (1997). Three-Dimensional Geometry and Topology. Vol. 1, ed. S. Levy Princeton Mathematical Series, Vol. 35, Princeton, NJ: Princeton University Press. MR 1435975
  • Thurston, W. P. (1998), How to see 3-Manifolds, Vol. 15, Topology of the Universe Conference, Cleveland, OH, 1997, pp. 2545–2571. MR 1649658
  • Trettel, S. Life in hyperbolic space, Available at: http://www.stevejtrettel.site/LifeInHyperbolic.pdf, 2018.
  • Troyanov, M. (1998). L’horizon de Sol, Expos. Math. 16(5):441–479. MR 1656902
  • Velho, L., da Silva, V., Novello, T. (2020). Immersive visualization of the classical non-euclidean spaces using real-time ray tracing in VR, Proceedings of Graphics Interface 2020, GI 2020, Canadian Human-Computer Communications Society/Société canadienne du dialogue humain-machine, pp. 423–430.
  • Weeks, J. (xxxx). Curved Spaces, a flight simulator for multiconnected universes. Available at: http://www.geometrygames.org/CurvedSpaces/.
  • Weeks, J. (2020). Virtual reality simulations of curved spaces, arXiv:2011.00510.
  • Weeks, J. (2002). Real-time rendering in curved spaces. IEEE Comp. Graph. Appl. 22(6):90–99.
  • Wong, J. (xxxx). Ray marching and signed distance functions. Available at: http://jamie-wong.com/2016/07/15/ray-marching-signed-distance-functions/.

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