Programming assignment

1. Programming assignment • You need to implement the following: • Display basic mesh Information • Find the number of vertices, edges, faces, boundaries, and compute Euler characteristic and number of genus • Corresponding to function Mesh::DisplayMeshInfo() • Compute normal at each vertex • Corresponding to function Mesh::ComputeVertexNormals() • Compute mean curvature at each vertex • Corresponding to function Mesh::ComputeVertexCurvatures()

2. Programming assignment • Explicit umbrella smoothing • Corresponding to function Mesh::UmbrellaSmooth () • Implicit umbrella smoothing • Corresponding to function Mesh::ImplicitUmbrellaSmooth ()

3. Wavefront OBJ file format comment coordinates of vertex 1 three vertices of each face counter-clockwise order as you look at the face from outside v2 v1 v4 v3

4. How to modify program arguments 1) Select the menu item “Project  Property”

5. How to modify program arguments 2) Select the “Debugging” tag 3) Type your arguments here 4) Press enter to comfirm

6. Normal at a vertex 0 t1 • From Siggraph 2000 subdivision course notes, the normal vector at an interior vertex of valence k can be computed as t1 x t2, where ti are tangent vectors computed as: • Example: for valence four, the masks are [1, 0, –1, 0] and [0, 1, 0, –1]. 1 -1 0 1 t2 0 0 -1 Masks for valence four

7. Normal at a vertex • At a boundary vertex p with valence k, the normal is computed as talong x tacross computed as follows: pk-1 p p1 p0

8. Mean curvature at a vertex • From Siggraph99 Desbrun et al , the discrete mean curvature at an interior vertex p with valence k can be computed as the L2-norm of p pj where A is the sum of areas of all the triangles sharing the vertex p p pj-1 one term of the summation, corresponding to edge p pj pj Pj+1

9. Explicit Umbrella Smoothing • Fairing operator • In matrix form • In your implementation you do not need to build the matrix Drawbacks: small time step for large mesh  slow Where t is time stamp

10. Implicit Umbrella Smoothing • Explicit updating • Allows large time step • In your implementation setting to 1 is okay • You can use biconjugate gradient (BCG) method to solve the linear system You need to solve a sparse linear system

11. Sparse Linear System • You need to build the sparse linear system by create a Matrix object • Use Matrix::AddElement to add an element • Use Matrix::SortMatrix to sort the elements after adding all of them • You need to implement the function Matrix::BCG() to solve the linear system • Use Matrix::Multiply to compute b = Ax • Use Matrix::PreMultiply to compute bT = xTA

12. Using Boundary Half-edge • For open meshes, you can also maintain boundary half-edges such that the searching and updating will be much more easy. (every things are circular list, also no need to handle NULL pointers) • Here the boundary half-edges are shown in red as the triangles are being loaded.

13. Useful Classes and Functions • Classes OneRingHEdgeand OneRingVertex provide an interface to access the one-ring neighboring half-edges and vertices of a given vertex. • Function Vertex::Valence() let you know the valence (# of neighboring vertices) of a vertex.

14. Rendering in OpenGL (Flat) One normal for a triangle

15. Rendering in OpenGL (Smooth) One normal for each vertex