Volumetric Quasi-conformal Mappings - Quasi-conformal Mappings for Volume Deformation with Applications to Geometric Modeling

Alexander Naitsat, Emil Saucan, Yehoshua Y. Zeevi

2015

Abstract

Due to intrinsic differences between surfaces and higher dimensional objects, some important results regarding surfaces can not be extended to volumetric domains. Most significantly, there exist no conformal volumetric maps apart from Möbius transformations. Although it is sometime stated explicitly, it is often overlooked that existing methods of volume parameterization produce only quasi-conformal maps, which may be “far from conformality”. We therefore introduce methods for assessing the extent of the local and global volumetric deformation by means of the amount of conformal distortion produced. To this end we first illustrate basic three-dimensional quasi-conformal deformations that are produced by parameterization techniques, and highlight theoretical issues associated with spatial quasi-conformal mappings, and the relation that exists between the geometry of the domain and conformal distortion.

References

  1. Ahlfors, L. V. (2006). Lectures on quasiconformal mappings. Providence, RI: American Mathematical Society (AMS), 2nd enlarged and revised edition.
  2. Almgren, F. J. and Rivin, I. (1999). The mean curvature integral is invariant under bending. In The David Epstein 60th birthday Festschrift. International Press.
  3. Amanatides, J. and Woo, A. (1987). A fast voxel traversal algorithm for ray tracing. In Eurographics 7887, pages 3-10. Elsevier Science Publishers, Amsterdam, North-Holland.
  4. Caraman, P. (1974). n-dimensional quasiconformal (QCF) mappings. Revised, enlarged and translated from the Roumanian by the author.
  5. Dukowicz, J. (1988). Efficient volume computation for three-dimensional hexahedral cells. Journal of Computational Physics.
  6. Karabassi, E.-A., Papaioannou, G., and Theoharis, T. (1999). A fast depth-buffer-based voxelization algorithm. Journal of Graphics Tools: JGT, 4(4):5-10.
  7. Lev, R., Saucan, E., and Elber, G. (2007). Curvature estimation over smooth polygonal meshes using the half tube formula. In Martin, R. R., Sabin, M. A., and Winkler, J. R., editors, IMA Conference on the Mathematics of Surfaces, volume 4647 of Lecture Notes in Computer Science, pages 275-289. Springer.
  8. Perreault, S. (2007). Octree C++ Class Template. http://nomis80.org/code/octree.html.
  9. Rickman, S. (1993). Quasiregular mappings, chapter 1, page 15. Berlin: Springer-Verlag.
  10. Si, H. (2009). A Quality Tetrahedral Mesh Generator and Three-Dimensional Delaunay Triangulator. Tetgen, http://tetgen.berlios.de.
  11. Väisälä, J. (1971). Lectures on n-Dimensional Quasiconformal Mappings. Springer-Verlag Berlin Heidelberg New York.
  12. van Oosterom A, S. J. (1983). The solid angle of a plane triangle. In IEEE Trans Biomed Eng.
  13. Wang, Y., Gu, X., Yau, S.-T., et al. (2003). Volumetric harmonic map. In Communications in Information & Systems, volume 3, pages 191-202. International Press of Boston.
  14. Willmore, T. (1993). Riemannian geometry. Oxford: Clarendon Press.
  15. Xia, J., He, Y., Han, S., Fu, C.-W., Luo, F., and Gu, X. (2010). Parameterization of star-shaped volumes using green's functions. In Mourrain, B., Schaefer, S., and Xu, G., editors, GMP, volume 6130 of Lecture Notes in Computer Science, pages 219-235. Springer.
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Paper Citation


in Harvard Style

Naitsat A., Saucan E. and Zeevi Y. (2015). Volumetric Quasi-conformal Mappings - Quasi-conformal Mappings for Volume Deformation with Applications to Geometric Modeling . In Proceedings of the 10th International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2015) ISBN 978-989-758-087-1, pages 46-57. DOI: 10.5220/0005298900460057


in Bibtex Style

@conference{grapp15,
author={Alexander Naitsat and Emil Saucan and Yehoshua Y. Zeevi},
title={Volumetric Quasi-conformal Mappings - Quasi-conformal Mappings for Volume Deformation with Applications to Geometric Modeling},
booktitle={Proceedings of the 10th International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2015)},
year={2015},
pages={46-57},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0005298900460057},
isbn={978-989-758-087-1},
}


in EndNote Style

TY - CONF
JO - Proceedings of the 10th International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, (VISIGRAPP 2015)
TI - Volumetric Quasi-conformal Mappings - Quasi-conformal Mappings for Volume Deformation with Applications to Geometric Modeling
SN - 978-989-758-087-1
AU - Naitsat A.
AU - Saucan E.
AU - Zeevi Y.
PY - 2015
SP - 46
EP - 57
DO - 10.5220/0005298900460057