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Michael S.Floater G éza Kós Martin Reimers CAGD 22(2005) 623-631 Reporter: Zhang Xingwang

Mean Value Coordinates in 3D. Michael S.Floater G éza Kós Martin Reimers CAGD 22(2005) 623-631 Reporter: Zhang Xingwang. Overview. 1. About the authors. 2. Motivation. 3. Introduction. 4. Mean value coordinates in 3D. 5. Convex polyhedra. 6. Numerical examples.

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Michael S.Floater G éza Kós Martin Reimers CAGD 22(2005) 623-631 Reporter: Zhang Xingwang

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  1. Mean Value Coordinates in 3D Michael S.Floater Géza Kós Martin Reimers CAGD 22(2005) 623-631 Reporter: Zhang Xingwang

  2. Overview 1. About the authors 2. Motivation 3. Introduction 4. Mean value coordinates in 3D 5. Convex polyhedra 6. Numerical examples 7. Conclusions and future work

  3. About the Authors • Michael S.Floater: • Department of Informatics of the University of Oslo, Centre of Mathematics for Applications(CMA) • Geometric modeling, Numerical analysis, Approximation theory • Géza Kós • Department of Analysis at Eötvös University in Budapest, Hungary • Approximation theory, Surface and solid modeling, Surface reconstruction. • Martin Reimers • Postdoc at CMA, University of Oslo • Geometric modeling&splines, Approximation theory, Mesh based modeling, Computer graphics

  4. Motivation A point represented as a convex combination of its neighboring vertices Key: barycentric coordinates Generalizing coordinates to convex polyhedra and the kernels of star-shaped polyhedra

  5. Introduction • Mean value coordinates in 2D(Floater 2003) • Applications: • Convex combination maps between pairs of planar regions (Surazhsky and Gotsman, 2003). • Smoothly interpolating piecewise linear height data given on the boundary of a convex polygon.

  6. Mean Value Coordinates in Some notations:

  7. Mean Value Coordinates in Some notations:

  8. Mean Value Coordinates in Tetrahedron

  9. Mean Value Coordinates in Spherical triangle

  10. Mean Value Coordinates in

  11. Mean Value Coordinates in

  12. Mean Value Coordinates in See

  13. Convex Polyhedra

  14. Convex Polyhedra

  15. Numerical Examples

  16. Numerical Examples

  17. Numerical Examples

  18. Numerical Examples

  19. Conclusions and Future Work • Natural extension of mean value coordinates to kernels of star-shaped polyhedra • 3D coordinates well-defined everywhere in a convex polyhedron, including the boundary • Polyhedron with multi-sided facets, first triangulate each facet. Depending on the choice of triangulation. • Extend 3D coordinates to arbitrary points, even for arbitrary polyhedra.

  20. Any Questions?

  21. Thanks for you attention!

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