COMPUTER GRAPHICS CHAPTER 8 CS 482 – Fall 2017 TRIANGLE MESHES 3D MESHES MESH OPERATIONS
3D MESHES TRIANGLES VERSUS QUADRILATERALS 3D shapes are usually represented in graphics systems with polygonal meshes. Triangle meshes and quadrilateral meshes are most commonly used. Triangle Mesh Advantages Quad Mesh Advantages Triangles are always planar, quads are not It’s easier to model the contours of a 3D surface with quads It’s easier to tile a surface with triangles of a uniform size It’s easier to map textures to quads than to triangles CS 482 – Fall 2017 CHAPTER 8: TRIANGLE MESHES PAGE 76
3D MESHES MANIFOLD MESHES A mesh is a manifold if it satisfies the following two conditions: Each edge in the mesh is incident with only one or two faces The faces incident to each vertex form either an open or a closed fan Open Fan Closed Fan CS 482 – Fall 2017 CHAPTER 8: TRIANGLE MESHES PAGE 77
3D MESHES TRIANGLE ORIENTATION Triangle manifold meshes lend themselves to orientation, which facilitates the determination of consistent references to the “inside” and “outside” of polyhedral surfaces. Start by ordering the vertices of any triangle in the mesh. For each shared edge, order the vertices of each adjacent triangle so the shared edge’s vertices are in the opposite order. Continue, one adjacent triangle at a time, until the entire mesh has been oriented. CS 482 – Fall 2017 CHAPTER 8: TRIANGLE MESHES PAGE 78
3D MESHES TRIANGLE ORIENTATION EXAMPLE Note that the counterclockwise orientation of each triangle yields normal vectors (via the cross product) that all point exterior to the polyhedron. CS 482 – Fall 2017 CHAPTER 8: TRIANGLE MESHES PAGE 79
CHAPTER 8: TRIANGLE MESHES 3D MESHES STORAGE REQUIREMENTS One of the most common means of storing a 3d triangle mesh is the vertex-table-and triangle-table approach. TRIANGLES Vertex Index T1 2 3 1 T2 4 T3 5 T4 6 … VERTICES Coordinates 1 (2.4,3.1,2.1) 2 (2.1,2.6.2.4) 3 (2.6,2.8,2.5) 4 (2.7,3.1,2.8) 5 (2.0,2.6,2.3) 6 (2.4,2.9,2.5) … With T triangles, E edges, and V vertices, and assuming 4 bytes per integer and 4 bytes per floating point number, this yields a total of 12T+12V bytes for storing the entire structure. For a convex polyhedron, E = 1.5T (since each triangle has three edges and every edge is shared by two triangles), and V-E+T= 2 (by Euler’s Formula), so the memory requirement is 12T+12V = 18T+24 (= 36V-48) bytes. CS 482 – Fall 2017 CHAPTER 8: TRIANGLE MESHES PAGE 80
MESH OPERATIONS EDGE COLLAPSE Mesh simplification is a class of algorithms that transforms a mesh into one with fewer vertices, edges, and faces, reducing its complexity while preserving certain properties (e.g., geometric distance, visual appearance). A B With edge collapse, an edge is removed and its endpoints are merged into a single vertex. The determination of the merged vertex’s position may significantly impact the local appearance of the simplified mesh. A B M Merging to one endpoint Merging to the other endpoint Merging to the midpoint CS 482 – Fall 2017 CHAPTER 8: TRIANGLE MESHES PAGE 81
MESH OPERATIONS REPEATED EDGE COLLAPSE IN THIS EXAMPLE, THE SHORTEST EDGES IN THE POLYHEDRAL OBJECT ARE COLLAPSED, REDUCING THE NUMBER OF FACES BY ONE THOUSAND WITH EACH STEP. ORIGINAL: 4000 FACES REDUCED TO 3000 FACES REDUCED TO 2000 FACES REDUCED TO 1000 FACES CS 482 – Fall 2017 CHAPTER 8: TRIANGLE MESHES PAGE 82
CHAPTER 8: TRIANGLE MESHES MESH OPERATIONS EDGE SWAP Mesh beautification is a class of algorithms that transforms a triangular mesh into one with near-equilateral triangular faces and/or a near uniform distribution of vertices, in order to eliminate irregularities in the appearance of the original mesh. Aspect Ratio: 11 Determine the triangular face with the largest aspect ratio (the circumscribed rectangular bounding box with the largest length-to-width ratio) Determine the largest edge of that triangle and find the other triangular face that shares that edge Swap that edge with the other edge that serves as a diagonal for that two-triangle quadrilateral CS 482 – Fall 2017 CHAPTER 8: TRIANGLE MESHES PAGE 83