Most of the projects below come with some papers. I can send you the references if you are interested.

Programming projects:

1. Surface Triangulation
• Given a surface in a non-mesh representation (e.g. implicit functions or parametric patches), triangulate the surface with quality triangles.
• Moreover, maintain the triangulation when the surface is deforming
2. Shape analysis and comparison
• Given a lot of shapes, align, compare and differentiate
3. Skin surface topics
• 2D Skin visualization (Java, C++, flash, etc.)
• 3D Skin triangulation and deformation
• Bezier triangle decomposition for skin surfaces.
• Skin Deformation in R² or R³
• Measurement derivatives computation.
• The volume and surface area of a union of balls are done. Next, derive the volume/surface area/derivatives for skin surfaces. More precisely, how to compute the surface area, volume and their derivatives of sphere and hyperboloid patches in the mixed cells.
• Deformation from a simplicial complex to its skin approximation.
• We got a paper on approximation polygonal objects with skin surfaces. Only a little gap is left, namely, deforming the objects to the very close skin. This can be paper work also.
4. Betti numbers computation in R⁴.
• They had been computed in 3 dimensions. Moving up one dimension maybe not that difficult. That is how to compute how many components, cycles, tunnels and voids in a 4 dimensional simplicial complex.
5. Volume meshing. with skin boundary
6. Shape Registration
7. Pocket extraction
8. Concepts visualization.
• To write some animation or tools to help the illustration of abstract ideas in our course.
9. 4-coloring a planar graph.
• I have a (new?) conjecture on coloring a planar graph by 3-coloring of the edges in O(n²) time. I would like to try it out.
10. Alpha shapes, weighted triangulation data structure R³.
• To write code for the basic triangle structure (mentioned in the lecture), with one application of your choice.
11. Decomposing a polygon in O(n) time.
12. Weighted Deluanay complex computation in R^d.
• Construct the Delaunay complexes for any dimension by the randomized incremental algorithm.
13. Molecular Modeling.
• Model some molecules from the protein data bank by the skin surface. Analyze the features and chemical properties of the molecules based on the shapes. One of the possibility is, we already have the tetrahedral meshes of the skin body. It will be good if we solve the Poisson-Boltzmann Equation for the electrostatics potential computation.
14. Medial axis approximation.
• Compute the medial axis of the mesh approximating the skin surface.

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Free projects. You can propose some projects by yourself.

Papers

1. Surveys or attempts to open problems