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**Mean Value Coordinates for Closed Triangular Meshes**

Tao Ju Scott Schaefer Joe Warren

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**Overview About the authors About the work Previous works**

MVC for closed polygons MVC for closed meshes Algorithm Applications Conclusions and future work

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About author Tao Ju: M.S. and Ph.D Rice UniversityAdvisor, Dr. Joe Warren B.S. Tsinghua University Research interests lie in the field of computer graphics and its applications in bio-medical research. Scott Schaefer: M.S. Computer Science Rice University 2003 Currently a Ph.D B.S. Computer Science and Mathematics Trinity University Research interests lie in the field of computer graphics Joe Warren: Department of Computer Science6100 South MainRice niversity Research interests are centered around the general problem of representing geometric shape.

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**About the work Given a closed Triangular Mesh,**

construct a function that interpolates a set of values defined at vertices of the mesh. Parameterize the interior points of the mesh.

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Illustration

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**Previous works Let be points in the plane with arranged in an**

anticlockwise ordering Around The points form a star -shaped polygon with in its kernel. Our Aim is to study sets of weights such that

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Wachspress[75] Shortcoming:

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**Mean value coordinates[03]**

Mean value theorem for harmonic functions Mean value coordinates

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Example Pole(divsions by 0) No Pole

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**Mean value interpolation**

Discrete :

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**Mean value interpolation**

Continuous:

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Important properties

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**MVC for closed polygons**

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**MVC for closed polygons**

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**MVC for closed polygons**

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**MVC for closed meshes Symmetry:**

Project surface onto sphere centered at v m = mean vector (integral of unit normal over spherical triangle) Symmetry:

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**MVC for closed meshes Given spherical triangle, compute mean**

vector (integral of unit normal) Build wedge with face normals Apply Stokes’ Theorem,

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**MVC for closed meshes Compute mean vector: Calculate weights**

By Sum over all triangles

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Algorithm

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Robust algorithm

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Pseudo-code

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**Applications Boundary value interpolation Volumetric textures**

Surface Deformation

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**Boundary value interpolation**

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Volumetric textures

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Surface Deformation

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**Conclusions and Future work**

Mean value coordinates are a simple,but powerful method for creating functions that interpolate values assigned to vertices of a closed mesh. One important generalization would be to derive mean value coordinates for piecewise linear mesh with arbitrary closed polygons as faces.

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Thanks all!

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1 Planes, Polygons and Objects ©Anthony Steed 1999-2005.

1 Planes, Polygons and Objects ©Anthony Steed 1999-2005.

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