Efficient Ray Traversal of Constrained Delaunay Tetrahedralization

Maxime Maria, Sébastien Horna, Lilian Aveneau


Acceleration structures are mandatory for ray-tracing applications, allowing to cast a large number of rays per second. In 2008, Lagae and Dutr\'{e} have proposed to use Constrained Delaunay Tetrahedralization (CDT) as an acceleration structure for ray tracing. Our experiments show that their traversal algorithm is not suitable for GPU applications, mainly due to arithmetic errors. This article proposes a new CDT traversal algorithm. This new algorithm is more efficient than the previous ones: it uses less arithmetic operations; it does not add extra thread divergence since it uses a fixed number of operation; at last, it is robust with 32-bits floats, contrary to the previous traversal algorithms. Hence, it is the first method usable both on CPU and GPU.


  1. Aila, T. and Laine, S. (2009). Understanding the Efficiency of Ray Traversal on GPUs. In High-Performance Graphics, HPG 7809, pages 145-149.
  2. Aila, T., Laine, S., and Karras, T. (2012). Understanding the efficiency of ray traversal on GPUs - Kepler and Fermi addendum. Technical report, NVIDIA Corp.
  3. Bentley, J. L. (1975). Multidimensional binary search trees used for associative searching. Communications of the ACM, 18(9):509-517.
  4. Chew, L. P. (1989). Constrained Delaunay triangulations. Algorithmica, 4:97-108.
  5. Delaunay, B. (1934). Sur la sphère vide. Ì la mémoire de Georges Voronoï. Bulletin de l'Académie des Sciences de l'URSS, (6):793-800.
  6. Devillers, O. and Pion, S. (2003). Efficient Exact Geometric Predicates for Delaunay Triangulations. In 5th Workshop on Algorithm Engineering and Experiments, ALENEX 7803, pages 37-44.
  7. Edelsbrunner, H. and Tan, T. S. (1992). An upper bound for conforming delaunay triangulations. In 8th Annual Symposium on Computational Geometry, SCG 7892, pages 53-62.
  8. Ernst, M. and Greiner, G. (2007). Early Split Clipping for Bounding Volume Hierarchies. In IEEE Symposium on Interactive Ray Tracing, RT 7807, pages 73-78.
  9. Foley, T. and Sugerman, J. (2005). KD-tree Acceleration Structures for a GPU Raytracer. In ACM SIGGRAPH/EUROGRAPHICS conference on Graphics Hardware, HWWS 7805, pages 15-22.
  10. Fortune, S. (1999). Topological Beam Tracing. In 15th Annual Symposium on Computational Geometry, SCG 7899, pages 59-68.
  11. Fujimoto, A., Tanaka, T., and Iwata, K. (1986). ARTS: Accelerated Ray-Tracing System. IEEE Computer Graphics and Applications, 6(4):16-26.
  12. Garrity, M. P. (1990). Raytracing Irregular Volume Data. ACM SIGGRAPH Computer Graphics, 24(5):35-40.
  13. Günther, J., Popov, S., Seidel, H.-P., and Slusallek, P. (2007). Realtime Ray Tracing on GPU with BVHbased Packet Traversal. In IEEE Symposium on Interactive Ray Tracing 2007, RT 7807, pages 113-118.
  14. Havran, V. (2000). Heuristic Ray Shooting Algorithms. PhD thesis, Department of Computer Science and Engineering, Faculty of Electrical Engineering, Czech Technical University in Prague.
  15. Kalojanov, J., Billeter, M., and Slusallek, P. (2011). TwoLevel Grids for Ray Tracing on GPUs. Computer Graphics Forum, 30(2):307-314.
  16. Kay, T. L. and Kajiya, J. T. (1986). Ray Tracing Complex Scenes. ACM SIGGRAPH Computer Graphics, 20(4):269-278.
  17. Lagae, A. and Dutré, P. (2008). Accelerating Ray Tracing using Constrained Tetrahedralizations. Computer Graphics Forum, (4):1303-1312.
  18. MacDonald, D. J. and Booth, K. S. (1990). Heuristics for Ray Tracing Using Space Subdivision. The Visual Computer, 6(3):153-166.
  19. Mahovsky, J. and Wyvill, B. (2006). Memory-conserving bounding volume hierarchies with coherent raytracing. Computer Graphics Forum, 25(2):173-182.
  20. Maria, M., Horna, S., and Aveneau, L. (2017). Constrained Convex Space Partition for Ray Tracing in Architectural Environments. Computer Graphics Forum.
  21. Marmitt, G. and Slusallek, P. (2006). Fast Ray Traversal of Tetrahedral and Hexahedral Meshes for Direct Volume Rendering. In 8th Joint EG / IEEE VGTC Conference on Visualization, EUROVIS 7806, pages 235-242.
  22. Miller, G. L., Talmor, D., Teng, S.-H., Walkington, N., and Wang, H. (1996). Control Volume Meshes using Sphere Packing: Generation, Refinement and Coarsening. In 5th International Meshing Roundtable, IMR 7896, pages 47-62.
  23. Platis, N. and Theoharis, T. (2003). Fast Ray-Tetrahedron Intersection Using Plucker Coordinates. Journal of Graphics Tools, 8(4):37-48.
  24. Purcell, T. J., Buck, I., Mark, W. R., and Hanrahan, P. (2002). Ray Tracing on Programmable Graphics Hardware. ACM Transactions on Graphics, 21(3):703-712.
  25. Reshetov, A., Soupikov, A., and Hurley, J. (2005). Multilevel Ray Tracing Algorithm. ACM Transactions on Graphics, 24(3):1176-1185.
  26. Rubin, S. M. and Whitted, T. (1980). A 3-dimensional representation for fast rendering of complex scenes. ACM SIGGRAPH Computer Graphics, 14(3):110-116.
  27. Shewchuk, J. R. (1996). Adaptive precision floating-point arithmetic and fast robust geometric predicates. Discrete & Computational Geometry, 18:305-363.
  28. Shewchuk, J. R. (1998). Tetrahedral Mesh Generation by Delaunay Refinement. In 14th Annual Symposium on Computational Geometry, SCG 7898, pages 86-95.
  29. Shoemake, K. (1998). Plücker coordinate tutorial. Ray Tracing News, 11:20-25.
  30. Si, H. (2006). On Refinement of Constrained Delaunay Tetrahedralizations. In 15th International Meshing Roundtable, IMR 7806, pages 509-528.
  31. Si, H. (2015). TetGen, a Delaunay-Based Quality Tetrahedral Mesh Generator. ACM Transactions on Mathematical Software, 41(2).
  32. Wald, I., Slusallek, P., Benthin, C., and Wagner, M. (2001). Interactive Rendering with Coherent Ray Tracing. Computer Graphics Forum, 20(3):153-165.
  33. Zhou, Q., Grinspun, E., Zorin, D., and Jacobson, A. (2016). Mesh Arrangements for Solid Geometry. ACM Transactions on Graphics, 35(4).

Paper Citation

in Harvard Style

Maria M., Horna S. and Aveneau L. (2017). Efficient Ray Traversal of Constrained Delaunay Tetrahedralization . In Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2017) ISBN 978-989-758-224-0, pages 236-243. DOI: 10.5220/0006131002360243

in Bibtex Style

author={Maxime Maria and Sébastien Horna and Lilian Aveneau},
title={Efficient Ray Traversal of Constrained Delaunay Tetrahedralization},
booktitle={Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2017)},

in EndNote Style

JO - Proceedings of the 12th International Joint Conference on Computer Vision, Imaging and Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2017)
TI - Efficient Ray Traversal of Constrained Delaunay Tetrahedralization
SN - 978-989-758-224-0
AU - Maria M.
AU - Horna S.
AU - Aveneau L.
PY - 2017
SP - 236
EP - 243
DO - 10.5220/0006131002360243