PA1 Supplementary notes 1 Programming assignment You need to implement the following: 1. 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() 2. Compute normal at each vertex Corresponding to function Mesh::ComputeVertexNormals() 3. Compute mean curvature at each vertex Corresponding to function Mesh:: ComputeVertexCurvatures ()

PA1 Supplementary notes 2 Programming assignment 4.Explicit umbrella smoothing Corresponding to function Mesh::UmbrellaSmooth () Implicit umbrella smoothing Corresponding to function Mesh::ImplicitUmbrellaSmooth ()

PA1 Supplementary notes 3 Wavefront OBJ file format  comment  three vertices of each face  coordinates of vertex 1 v1 v2 v3 v4  counter-clockwise order as you look at the face from outside

PA1 Supplementary notes 4 How to modify program arguments 1) Select the menu item “Project  Property”

PA1 Supplementary notes 5 How to modify program arguments 2) Select the “Debugging” tag 3) Type your arguments here 4) Press enter to comfirm

PA1 Supplementary notes 6 Normal at a vertex From Siggraph 2000 subdivision course notes, the normal vector at an interior vertex of valence k can be computed as t 1 x t 2, where t i are tangent vectors computed as: Example: for valence four, the masks are [1, 0, –1, 0] and [0, 1, 0, –1] t1 t2 Masks for valence four

PA1 Supplementary notes 7 Normal at a vertex At a boundary vertex p with valence k, the normal is computed as t along x t across computed as follows: p0p0 p1p1 p k-1 p

PA1 Supplementary notes 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 where A is the sum of areas of all the triangles sharing the vertex p p pjpj p p j-1 pjpj P j+1 one term of the summation, corresponding to edge p p j

PA1 Supplementary notes 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

PA1 Supplementary notes 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

PA1 Supplementary notes 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 b T = x T A

PA1 Supplementary notes 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.

PA1 Supplementary notes 13 Useful Classes and Functions Classes OneRingHEdge and 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.

PA1 Supplementary notes 14 Rendering in OpenGL (Flat) One normal for a triangle

PA1 Supplementary notes 15 Rendering in OpenGL (Smooth) One normal for each vertex