Polygonal Mesh. Geometric object made of vertices, edges and faces. Faces are polygons. Polyhedron. Triangular mesh Quad mesh. Pyramid Cube Sphere (?

Transcription

1 1 Mesh Modeling

2 Polygonal Mesh Geometric object made of vertices, edges and faces Polyhedron Pyramid Cube Sphere (?) Can also be 2D (although much less interesting) Faces are polygons Triangular mesh Quad mesh Mesh Modeling- Center for Graphics and Geometric Computing, Technion 2

3 Vertex a point in space Polygonal Mesh Can include more info like color and normal Edge a segment between two vertices Face a polygon defined by a set of vertices and edges Mesh Modeling- Center for Graphics and Geometric Computing, Technion 3

4 Data Structures Why do we need a data structure? To store the mesh To make queries Which vertices are connected to vertex v? What are the neighbors of face f? Geometric operation Add vertices Split edges Mesh Modeling- Center for Graphics and Geometric Computing, Technion 4

5 Face list Lists of coordinates Polygons are unrelated Polygon Soup v 3 v 6 face vertices (ccw) f 1 (v 1, v 2, v 3 ) (v 2, v 4, v 3 ) f 3 f v 1 f 1 f 2 f 4 v 4 v 2 v 5 Mesh Modeling- Center for Graphics and Geometric Computing, Technion 5

6 Vertex list Vertex coordinates Lists of connected vertices Vertex-Vertex v 3 v 6 vertex coordinates Neighbors v 1 (x 1, y 1, z 1 ) (v 2, v 3 ) (v 1, v 3, v 4 ) v 1 v 4 v 2 (x 2, y 2, z 2 ) v 3 (x 3, y 3, z 3 ) (v 1, v 2, v 4, v 6 ) v 2 v 5 Edges and faces are implicit Mesh Modeling- Center for Graphics and Geometric Computing, Technion 6

7 Vertex list Vertex coordinates Vertex-Vertex Lists of connected vertices Adjacency matrix v 3 v 6 vertex coordinates v 1 (x 1, y 1, z 1 ) (x 2, y 2, z 2 ) v 1 v 2 v 4 v 5 Edges and faces are implicit v 3 (x 3, y 3, z 3 ) v 1 v 2 v 3 v v v Mesh Modeling- Center for Graphics and Geometric Computing, Technion 7

8 Face-Vertex Vertex list Vertex coordinates Lists of incident faces v 3 Face list Lists of incident vertices vertex coordinates v 6 v 1 (x1, y 1, z 1 ) (x 2, y 2, z 2 ) f 3 v 2 (x 3, y 3, z 3 ) v 1 f 1 f 2 v 4 f 4 v face vertices (ccw) f 1 (v 1, v 2, v 3 ) (v 2, v 4, v 3 ) v 2 v 5 f Mesh Modeling- Center for Graphics and Geometric Computing, Technion 8

9 v 1 Vertex list Vertex coordinates Lists of incident edges Edge list e 1 List of incident vertices, edges and faces v 3 e 3 f 1 f 2 v 2 e 5 e 2 e 4 Winged-Edge e 6 f 3 v 4 e 7 f 4 v 6 Face list vertex) v 5 Vertex coordinates Lists of incident edges coordinates Mesh Modeling- Center for Graphics and Geometric Computing, Technion 9 Edges v 1 (x 1, y 1, z 1 ) (e 1, e 2 ) v 2 (x 2, y 2, z 2 ) (e 2, e 3 ) v 3 (x 3, y 3, z 3 ) (e 1, e 5 ) face vertices Edges (ccw) f 1 (v 1, v 2, v 3 ) (e 1, e 2, e 3 ) f 2 (v 2, v 4, v 3 ) (v 3, v 4, v 5 )......

10 Winged-Edge Vertex list Vertex coordinates Lists of incident edges Edge list List of incident vertices, edges and faces v e 6 v 3 6 e 1 e 5 f 3 Face list Edge Vertex coordinates Lists of incident edges Vertices Edges Faces e 1 (v 1, v 3 ) (e 2, e 3 ) (f 1 ) e 2 (v 1, v 2 ) (e 1, e 3 ) (f 1 ) e 3 (v 2, v 3 ) (e 1, e 2, e 4, e 5 ) (f 1, f 2 ) v 1 e 3 f 1 f 2 e 2 e 4 v 2 v 4 e 7 f 4 v 5 Mesh Modeling- Center for Graphics and Geometric Computing, Technion 10

11 Normal Estimation Normals are not always available Face normals can always be computed Vertex normals are not even defined No correct answer. Estimate vertex normals as if shape was smooth Vertex normal is average of incident face normals What if some faces are larger than others? Vertex normal is weighted average of incident face normals Weights are face areas Mesh Modeling- Center for Graphics and Geometric Computing, Technion 11

12 Smoothing Move pointy vertices to make the mesh smooth New vertex position is average of neighbors positions Side effect Mesh shrinkage What will happen after infinite iterations? Mesh Modeling- Center for Graphics and Geometric Computing, Technion 12

13 Basic Operations Each element can be individually transformed Also groups of elements Moving vertices Mesh Modeling- Center for Graphics and Geometric Computing, Technion 13

14 Basic Operations Each element can be individually transformed Also groups of elements Moving edges Mesh Modeling- Center for Graphics and Geometric Computing, Technion 14

15 Basic Operations Each element can be individually transformed Also groups of elements Moving faces Mesh Modeling- Center for Graphics and Geometric Computing, Technion 15

16 Soft Selection Selelct a vertex, It s vicinity gets partially selected When the vertex is moved the vicinity makes a partial move Mesh Modeling- Center for Graphics and Geometric Computing, Technion 16

17 Soft Selection Selelct a vertex, It s vicinity gets partially selected When the vertex is moved the vicinity makes a partial move Mesh Modeling- Center for Graphics and Geometric Computing, Technion 17

18 Space Warp Non-linear transformation Each vertex is mapped to a different location For example f(x,y,z)=(cos(x+z),sin(x+z),z) Mesh Modeling- Center for Graphics and Geometric Computing, Technion 18

19 Space Warp Bend Mesh Modeling- Center for Graphics and Geometric Computing, Technion 19

20 Space Warp Twist Mesh Modeling- Center for Graphics and Geometric Computing, Technion 20

21 Space Warp Wave Mesh Modeling- Center for Graphics and Geometric Computing, Technion 21

22 Space Warp Deforming Buck Mesh Modeling- Center for Graphics and Geometric Computing, Technion 22

23 Edge Split Refining Meshes Create a new vertex dividing an edge Face Split Create a new edge dividing a face And many more Extrusion, Chamfer, Fillet Mesh Modeling- Center for Graphics and Geometric Computing, Technion 23