Seung-Wook Kim
Skip Abstract Section
Skip Supplemental Material SectionAbstract
A non-Euclidean space is characterized as a manifold with a specific structure that violates Euclid's postulates. This paper proposes to approximate a manifold with polytopes. Based on the scene designer's specification, the polytopes are automatically concatenated and embedded in a higher-dimensional Euclidean space. Then, the scene is navigated and rendered via novel methods tailored to concatenated polytopes. The proof-of-concept implementation and experiments with it show that the proposed methods bring the virtual-world users unusual and fascinating experiences, which cannot be provided in Euclidean-space applications.
Supplemental Material
- K Ayush and P Chaudhuri. 2017. Rendering curved spacetime in everyday scenes. In ACM SIGGRAPH 2017 Posters. 1--2.Google Scholar
Digital Library
- RG Baraniuk and MB Wakin. 2009. Random projections of smooth manifolds. Foundations of computational mathematics 9, 1 (2009), 51--77.Google Scholar
- DS Bernstein. 2009. Matrix mathematics. Princeton university press.Google Scholar
- J Blinn. 2003. Jim Blinn's corner: notation, notation, notation. Morgan Kaufmann.Google Scholar
- MT Bosch. 2020. N-dimensional rigid body dynamics. ACM Transactions on Graphics (TOG) 39, 4 (2020), 55--1.Google ScholarDigital Library
- M Cavallo. 2021. Higher Dimensional Graphics: Conceiving Worlds in Four Spatial Dimensions and Beyond. In Computer Graphics Forum, Vol. 40. Wiley Online Library, 51--63.Google Scholar
- CodeParade. 2018. Non-Euclidean Worlds Engine. Video. https://www.youtube.com/watch?v=kEB11PQ9Eo8. (Accessed January 27, 2022).Google Scholar
- R Coulon, EA Matsumoto, H Segerman, and S Trettel. 2020a. Non-euclidean virtual reality III: Nil. Proceedings of Bridges 2020: Mathematics, Music, Art, Architecture, Culture (2020), 153--160.Google Scholar
- R Coulon, EA Matsumoto, H Segerman, and S Trettel. 2020b. Non-euclidean virtual reality IV: Sol. Proceedings of Bridges 2020: Mathematics, Music, Art, Architecture, Culture (2020), 161--168.Google Scholar
- F Dassi, A Mola, and H Si. 2014. Curvature-adapted remeshing of CAD surfaces. Procedia Engineering 82 (2014), 253--265.Google ScholarCross Ref
- F Dassi, H Si, S Perotto, and T Streckenbach. 2015. Anisotropic finite element mesh adaptation via higher dimensional embedding. Procedia Engineering 124 (2015), 265--277.Google ScholarCross Ref
- Demruth. 2013. Antichamber. Video Game. (31 January 2013). http://www.antichamber-game.com/. Retrieved January 27, 2022.Google Scholar
- M Falk, T Schafhitzel, D Weiskopf, and T Ertl. 2007. Panorama maps with non-linear ray tracing. In Proceedings of the 5th international conference on Computer graphics and interactive techniques in Australia and Southeast Asia. 9--16.Google ScholarDigital Library
- D Harabor and A Grastien. 2011. Online graph pruning for pathfinding on grid maps. In Proceedings of the AAAI Conference on Artificial Intelligence, Vol. 25. 1114--1119.Google ScholarCross Ref
- PE Hart, NJ Nilsson, and B Raphael. 1968. A formal basis for the heuristic determination of minimum cost paths. IEEE transactions on Systems Science and Cybernetics 4, 2 (1968), 100--107.Google Scholar
- V Hart, A Hawksley, EA Matsumoto, and H Segerman. 2017a. Non-euclidean virtual reality I: explorations of H3, 2017., 33--40 pages.Google Scholar
- V Hart, A Hawksley, EA Matsumoto, and H Segerman. 2017b. Non-euclidean virtual reality II: explorations of H2 × E. Proceedings of Bridges 2017: Mathematics, Music, Art, Architecture, Culture (2017), 41--48.Google Scholar
- V Hart, A Hawksley, H Segerman, and MT Bosch. 2015. Hypernom: Mapping VR Headset Orientation to S3. Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture (2015), 387--390.Google Scholar
- B Lévy and N Bonneel. 2013. Variational anisotropic surface meshing with Voronoi parallel linear enumeration. In Proceedings of the 21st international meshing roundtable. Springer, 349--366.Google ScholarCross Ref
- A Macdonald. 2010. Linear and geometric algebra. CreateSpace Independent Publishing Platform.Google Scholar
- T Novello, V da Silva, and L Velho. 2020a. Global illumination of non-Euclidean spaces. Computers & Graphics 93 (2020), 61--70.Google ScholarCross Ref
- T Novello, V da Silva, and L Velho. 2020b. Visualization of Nil, Sol, and SL2 (R) geometries. Computers & Graphics 91 (2020), 219--231.Google ScholarCross Ref
- D Panozzo, E Puppo, M Tarini, and O Sorkine-Hornung. 2014. Frame fields: Anisotropic and non-orthogonal cross fields. ACM Transactions on Graphics (TOG) 33, 4 (2014), 1--11.Google ScholarDigital Library
- JR Silva et al. 2018. Ray Tracing in Non-Euclidean Spaces. Ph.D. Dissertation.Google Scholar
- Valve. 2007. Portal. Video Game. (10 October 2007). https://store.steampowered.com/app/400/Portal/. Retrieved January 27, 2022.Google Scholar
- N Verma. 2012. Distance preserving embeddings for general n-dimensional manifolds. In Conference on Learning Theory. JMLR Workshop and Conference Proceedings, 32--1.Google Scholar
- D Weiskopf, T Schafhitzel, and T Ertl. 2004. GPU-based nonlinear ray tracing. In Computer graphics forum, Vol. 23. Wiley Online Library, 625--633.Google Scholar
- H Whitney, J Eells, and D Toledo. 1992. Collected Papers of Hassler Whitney. Vol. 1. Nelson Thornes.Google Scholar
- Z Zhong, X Guo, W Wang, B Lévy, F Sun, Y Liu, W Mao, et al. 2013. Particle-based anisotropic surface meshing. ACM Trans. Graph. 32, 4 (2013), 99--1.Google ScholarDigital Library
- Z Zhong, W Wang, B Lévy, J Hua, and X Guo. 2018. Computing a high-dimensional euclidean embedding from an arbitrary smooth riemannian metric. ACM Transactions on Graphics (TOG) 37, 4 (2018), 1--16.Google ScholarDigital Library
Index Terms
Modeling and rendering non-euclidean spaces approximated with concatenated polytopes
Recommendations
Concatenated zigzag hadamard codes
In this correspondence, we introduce a new class of low-rate error correction codes called zigzag Hadamard (ZH) codes and their concatenation schemes. Each member of this class of codes is specified by a highly structured zigzag graph with each segment ...
Concatenated codes: serial and parallel
An analogy is examined between serially concatenated codes and parallel concatenations whose interleavers are described by bipartite graphs with good expanding properties. In particular, a modified expander code construction is shown to behave very much ...
Comments