1 Triangle Surfaces with Discrete Equivalence Classes Published in SIGGRAPH 2010 報告者 : 丁琨桓

2 Introduction Modeling freeform shapes has many uses modeling the body of a car for manufacturing purposes, or depicting the shape of a building for architectural applications.

3 Introduction the pieces that make up these shapes require a great deal of customization. This paper propose a technique that altered geometry such that each polygon falls into a set of discrete equivalence classes.

4 Discrete Equivalence Classes Input mesh Clustering Rigid Transformation Global Optimization output mesh Polygon Assignment & detect Canonical Triangles Mesh of Canonical Triangles Modified Geometry

5 Triangle Similarity Rigid transformation

6 Triangle Similarity (a 1,a 2,a 3 ) (b 1,b 2,b 3 ), (b 1,b 3,b 2 ) (b 2,b 1,b 3 ), (b 2,b 3,b 1 ) (b 3,b 1,b 2 ), (b 3,b 2,b 1 ) permutation

7 Triangle Similarity find the best Rigid Transformation Least-squares fitting of two 3-d point sets [Arun et al. 1987]. where, represent the polygon’s centroid, R is then given by R = UV T where M = UΣV T is the singular value decomposition of M

8 canonical triangle 5-Point Tensile Roof 1280 triangles

9 canonical triangle

10 canonical triangle

11 canonical triangle

12 Clustering Clustering begin our optimization with a single cluster Iteratively add a new cluster corresponding to the polygon with the worst error in the summation from similar Equation repeat this process until n clusters have been added

13 Clustering

14 Clustering 1280 triangles | 10 clusters canonical triangle

15 Clustering Before Global Optimization

16 Clustering Spacing between Triangles 20 clusters Before Global Optimization

17 Global Optimization Mesh Editing with Poisson-Based Gradient Field Manipulation [2004] Solve a Poisson equation to find the new positions of the vertices to match the canonical polygons. Disconnected Triangles

18 Global Optimization Wher α and β are small constants (0.001 and 0.01)

19 Global Optimization The Poisson equation attempts to find vertex positions for the shape P such that is minimized ∇ P i is the gradient of the triangle P i and △ i is the area of P i, C ind(i) is canonical polygon

20 Global Optimization (x i, n i ) is the closest point and normal on the initial shape P 0 to P i ’s centroid xixi nini

21 Global Optimization If p ℓ is a vertex on the boundary of P and y 1, y 2 are vertices on the boundary of P 0 such that their edge is the closest to p ℓ y1y1 y2y2 pℓpℓ

22 Global Optimization Before Global Optimization After Global Optimization

23 Clustering & Global Optimization 1-Clusters2-Clusters

24 Clustering & Global Optimization 3-Clusters4-Clusters

25 Clustering & Global Optimization 5-Clusters6-Clusters

26 Result

27 Result 1724 polygons optimized using 42 clusters