# Mean Value Coordinates for Closed Triangular Meshes

## Presentation on theme: "Mean Value Coordinates for Closed Triangular Meshes"— Presentation transcript:

Mean Value Coordinates for Closed Triangular Meshes
Tao Ju Scott Schaefer Joe Warren

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

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.

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.

Illustration

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

Wachspress[75] Shortcoming:

Mean value coordinates[03]
Mean value theorem for harmonic functions Mean value coordinates

Example Pole(divsions by 0) No Pole

Mean value interpolation
Discrete :

Mean value interpolation
Continuous:

Important properties

MVC for closed polygons

MVC for closed polygons

MVC for closed polygons

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

MVC for closed meshes Given spherical triangle, compute mean
vector (integral of unit normal) Build wedge with face normals Apply Stokes’ Theorem,

MVC for closed meshes Compute mean vector: Calculate weights
By Sum over all triangles

Algorithm

Robust algorithm

Pseudo-code

Applications Boundary value interpolation Volumetric textures
Surface Deformation

Boundary value interpolation

Volumetric textures

Surface Deformation

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.

Thanks all!