1. CS175 2003 1 CS 175 – Week 4 Triangle Mesh Smoothing Discrete Differential Geometry

2. CS175 2003 2 Overview triangle mesh basics data structures basic differential geometry discretization

3. CS175 2003 3 Triangle Mesh Basics triangle mesh consist of vertices edges triangles and it is an orientable manifold (with boundary)

4. CS175 2003 4 Triangle Mesh Basics terminology neighbours valency boundary Euler formula V – E + F = 2(1 – G) – B

5. CS175 2003 5 Data Structures extended indexed face set array of vertices coordinates index to one adjacent triangle array of triangles vertex indices indices of neighbouring triangles

6. CS175 2003 6 Data Structures directed half-edges array of vertices coordinates index to one adjacent half-edge array of half-edges vertex index neighbouring half-edge previous half-edge

7. CS175 2003 7 Differential Geometry Definition of tangent plane surface normal first & second fundamental form curvature Laplace Beltrami operator

8. CS175 2003 8 Discretization normal average adjacent triangle normals area or inverse area weighting average sphere normals …

9. CS175 2003 9 Discretization integral curvature values blending cylinders Gaussian curvature Gauss-Bonnet-Theorem s K = 2  –  i  i mean curvature s H = ¼  i ||e i ||  i