The Community for Technology Leaders
RSS Icon
Subscribe
Issue No.06 - November/December (2010 vol.16)
pp: 1311-1318
Olga Karpenko , UC Berkeley
Wilmot Li , Adobe Systems
Niloy Mitra , KAUST / IIT Delhi
Maneesh Agrawala , UC Berkeley
ABSTRACT
We present a technique for visualizing complicated mathematical surfaces that is inspired by hand-designed topological illustrations. Our approach generates exploded views that expose the internal structure of such a surface by partitioning it into parallel slices, which are separated from each other along a single linear explosion axis. Our contributions include a set of simple, prescriptive design rules for choosing an explosion axis and placing cutting planes, as well as automatic algorithms for applying these rules. First we analyze the input shape to select the explosion axis based on the detected rotational and reflective symmetries of the input model. We then partition the shape into slices that are designed to help viewers better understand how the shape of the surface and its cross-sections vary along the explosion axis. Our algorithms work directly on triangle meshes, and do not depend on any specific parameterization of the surface. We generate exploded views for a variety of mathematical surfaces using our system.
INDEX TERMS
exploded view diagrams, mathematical visualization, symmetry
CITATION
Olga Karpenko, Wilmot Li, Niloy Mitra, Maneesh Agrawala, "Exploded View Diagrams of Mathematical Surfaces", IEEE Transactions on Visualization & Computer Graphics, vol.16, no. 6, pp. 1311-1318, November/December 2010, doi:10.1109/TVCG.2010.151
REFERENCES
[1] Google 3D Warehouse.http://www.google.com/sketchup/3dwh.
[2] M. Agrawala, D. Phan, J. Heiser, J. Haymaker, J. Klingner, P. Hanrahan, and B. Tversky, Designing effective step-by-step assembly instructions. ACM Transactions on Graphics, 22 (3): 828–837, 2003.
[3] T. Banchoff, Beyond the third dimension : geometry, computer graphics, and higher dimensions. Scientific American Library : Distributed by W.H. Freeman, New York, 1990.
[4] W. V. Baxter, P. Barla, and K.-i. Anjyo, Compatible embedding for 2D shape animation. IEEE Transactions on Visualization and Computer Graphics, 15 (5): 867–879, 2009.
[5] P. Bourke, Geometry, surfaces, curves, polyhedra. http://local.wasp.uwa.edu.au/pbourke/geometry .
[6] S. Bruckner and M. E. Groller, Volumeshop: An interactive system for direct volume illustration. In Proceedings of IEEE Visualization 2005, pages 671–678, 2005.
[7] S. Bruckner and M. E. Groller, Exploded views for volume data. IEEE Transactions on Visualization and Computer Graphics, 12 (5): 1077–1084, 2006.
[8] C. D. Bruyns, S. Senger, A. Menon, K. Montgomery, S. Wildermuth, and R. Boyle, A survey of interactive mesh-cutting techniques and a new method for implementing generalized interactive mesh cutting using virtual tools. The Journal of Visualization and Computer Animation, 13 (1): 21–42, 2002.
[9] M. Burns and A. Finkelstein, Adaptive cutaways for comprehensible rendering of polygonal scenes. ACM Transactions on Graphics, 27 (5): 1–7, 2008.
[10] C. Curtis, A strange immersion of the torus in 3-space. website, 1992. http://www.otherthings.com/uwtorus.html.
[11] J. Diepstraten, D. Weiskopf, and T. Ertl, Interactive cutaway illustrations. In Computer Graphics Forum, pages 523–532, 2003.
[12] E. Driskill and E. Cohen, Interactive design, analysis, and illustration of assemblies. In SI3D `95: Proceedings of the 1995 symposium on Interactive 3D graphics, pages 27–34, 1995.
[13] S. Feiner and D. D. Seligmann, Cutaways and ghosting: satisfying visibility constraints in dynamic 3D illustrations. The Visual Computer, 8 (5&6): 292–302, 1992.
[14] G. Francis and J. M. Sullivan, Visualizing a sphere eversion. IEEE Transactions on Visualization and Computer Graphics, 10 (5): 509–515, 2004.
[15] G. K. Francis, A Topological Picturebook. Springer, 2 edition, 2007.
[16] W. Fulton, Algebraic curves. In Mathematics Lecture Note Series, 1974.
[17] E. Grinspun and A. Secord, Introduction to discrete differential geometry: the geometry of plane curves. In ACM SIGGRAPH 2005 Courses, 2005.
[18] D. Hoffman and J. T. Hoffman, The scientific graphics project. http://www.msri.org/about/sgp/jim/models index.html.
[19] S. Levy, D. Maxwell, and T. Munzner, Outside in (video), 1994.
[20] W. Li, M. Agrawala, B. Curless, and D. Salesin, Automated generation of interactive 3D exploded view diagrams. ACM Transactions on Graphics, 27 (3): 1–7, 2008.
[21] W. Li, M. Agrawala, and D. Salesin, Interactive image-based exploded view diagrams. In Proceedings of Graphics Interface 2004, pages 203–212, 2004.
[22] W. Li, L. Ritter, M. Agrawala, B., and D. Salesin, Interactive cutaway illustrations of complex 3D models. ACM Transactions on Graphics, 26 (3): 31, 2007.
[23] M. J. McGuffin, Sphere eversion program, 2005. http://www.dgp.toronto.edu/mjmcguff/eversion .
[24] M. J. McGuffin, L. Tancau, and R. Balakrishnan, Using deformations for browsing volumetric data. In Proc. of IEEE Visualization, page 53, 2003.
[25] N. J. Mitra, L. J. Guibas, and M. Pauly, Partial and approximate symmetry detection for 3D geometry. ACM Transactions on Graphics, 25 (3): 560–568, 2006.
[26] Y. Mori, S. Takahashi, T. Igarashi, Y. Takeshima, and I. Fujishiro, Automatic cross-sectioning based on topological volume skeletonization. In Smart Graphics, pages 175-184, 2005.
[27] C. Niederauer and M. Houston, M. Agrawala, and G. Humphreys, Non-invasive interactive visualization of dynamic architectural environments. In Proc. of the symp. on Interactive 3D graphics, pages 55–58, 2003.
[28] S. Owada, F. Nielsen, M. Okabe, and T. Igarashi, Volumetric illustration: designing 3D models with internal textures. ACM Transactions on Graphics, 23 (3): 322–328, 2004.
[29] A. Phillips, Turning a sphere inside out. Scientific American, 214: 112–120, 1966.
[30] M. Ruiz, I. Viola, I. Boada, S. Bruckner, M. Feixas, and M. Sbert, Similarity-based exploded views. In Proceedings of Symp. on Smart Graphics, pages 154–165, 2008.
[31] I. Viola, A. Kanitsar, and M. E. Groller, Importance-driven feature enhancement in volume visualization. IEEE Transactions on Visualization and Computer Graphics, 11 (4): 408–418, 2005.
[32] E. Weisstein, Eric weissteins world of mathematics. Wolfram Research,http:/mathworld.wolfram.com.
[33] X. Zabulis, J. Sporring, and S. C. Orphanoudakis, Perceptually relevant and piecewise linear matching of silhouettes. Pattern Recognition, 38 (1): 75–93, 2005.
17 ms
(Ver 2.0)

Marketing Automation Platform Marketing Automation Tool