On Triangle/Quad Subdivision Scott Schaefer and Joe Warren TOG 22(1) 28 – 36 , 2005

1. On Triangle/Quad SubdivisionScott Schaefer and Joe WarrenTOG22(1) 28–36, 2005 Reporter: Chen zhonggui 2005.10.27

2. About the authors • Scott Schaefer: • B.S in computer science and mathematics, Trinity University • M.S. in computer science, Rice University • Ph.D. candidate at Rice University • Research interests: computer graphics and computer-aided geometric design.

3. About the authors • Joe Warren: • Professor of computer science at Rice University • Associate editor of TOG • B.S. in computer science, math, and electrical engineering, Rice University • M.S. and Ph.D. in computer science, Cornell University • Research interests: subdivision, geometric modeling, and visualization.

4. Outline • Preview • Previous works • Catmull-Clark surface • Loop surface • Triangle/Quad Subdivision • On triangle/Quad Subdivision • Conclusion

5. Preview

6. Preview

7. Previous works • Chaikin, G.. An algorithm for high speed curve generation .Computer Graphics and Image Processing, 3(4):346-349, 1974 • E. Catmull and J. Clark. Recursively generated B-spline rurfaces on arbitrary topological meshes. Computer Aided Design, 10(6):350–355, 1978 • D. Doo and M. A. Sabin. Behaviour Of Recursive Subdivision Surfaces Near Extraordinary Points. Computer Aided Design, 10(6):356–360, 1978

8. Previous works • C. T. Loop. Smooth Subdivision Surfaces Based on Triangles.M.S. Thesis, departmentof Mathematics, University of tah, August 1987 • Stam, J., and Loop, C.. Quad/triangle subdivision. Comput. Graph. For. 22(1):1–7, 2003 • Levin, A. and Levin, D.. Analysis of quasi uniform subdivision. Applied Computat. Harmon. Analy. 15(1):18–32, 2003

9. Previous works • Warren, J., and Schaeffer, S.. A factored approach to subdivision surfaces. Comput. Graph. Applicat. 24:74-81, 2004 • Schaeffer, S., and Warren, J.. On triangle/quad subdivision. Transactions on Graphics. 24(1):28-36, 2005

10. Catmull-Clark SurfaceE. Catmull and J. Clark, 1978 New face point New edge point New vertex point Standard bicubic B-spline patch on a rectangular control-point mesh

11. Catmull-Clark Surface on Arbitrary Topology • Generalized subdivion rules: • New face point: the average of all he old points defining the face. • New edge point: the average of the midpoints of the old edge with the average of the new face points of the faces sharing the edge. • New vertex point: After one iteration Extraordinary vertex (not valence four vertex)

12. Factorization Step1. Linear subdivision Step2. Weighted averaging Averaging mask for regular vertex

13. Centroid averaging approach (b) Averaging the centroids (a) Computation of centroids

14. Subdivision Matrix V One-ring neighboring vertices of extraordinary vertex V M: a constant matrix

15. Property • continuous on the regular quad regions. • continuous at extraordinary vertices.

16. Loop SurfaceC. T. Loop, 1987 Extraordinary vertex (not valence six vertex) Original mesh Applying subdivision once

17. Loop Surface (1) Averaging mask for regular vertex (2) Averaging mask for extraordinary vertex ?

18. Centroid averaging approach (a) Centroid calculation for triangles (b) The result averaging mask

19. Property • continuous on the regular triangle regions. • continuous at extraordinary vertices but valence three vertices (valence three vertices are only ). Demo

20. Drawbacks of above surfaces • Catmull-Clark surfaces behave very poorly on triangle-only base meshes: A regular triangular mesh (left) behaves poorly with Catmull-Clark (middle) and behaves nicely with Loop.

21. Drawbacks of above surfaces • Loop schemes do not perform well on quad-only meshes. • Designers often want to preserve quad patches on regular areas of the surface where there are two “natural” directions. • It is often desirable to have surfaces that have a hybrid quad/triangle patch structure.

22. Triangle/Quad SubdivisionStam, J. and Loop, C., 2003 1. Initial shape 2. Linear subdivision 3. Weighted averaging

23. Averaging masks Averaging mask for regular quads Averaging mask for regular triangles

24. Averaging masks (a) Averaging masks for ordinary quad-triangles (b) Averaging mask for extraordinary vertex?

25. Weighted centroid averaging approach (a) Centroids are weighted by their angular contribution (b) The result averaging masks

26. Property • continuous on both the regular quad and the triangle regions of the mesh. • but not continuous at the irregular quad and triangle regions. • Cannot be along the quad/triangle boundary. Demo

27. On Triangle/Quad Subdivision The unified subdivision scheme • “Unzips” the mesh into disjoint pieces consisting of only triangles or only quads. (Levin and Levin [2003]) • Linear subdivision.(Stam and Loop [2003]) • Weighted average of centroids. (Warren and Schaefer [2004])

28. Unzipping pass • Identify edges on the surface contained by both triangles and quads. • Apply the unzipping masks ( , ) to this curve network. • Linear subdivision. • Weighted average of centroids

29. Property • continuous on both the regular quad and the triangle regions of the mesh. • continuous along the quad/triangle boundary. • continuous at the irregular quad and triangle regions.

30. Conclusion • We have presented a subdivision scheme for mixed triangle/quad surfaces that is everywhere except for isolated, extraordinary vertices. • The method is easy to code since it is a simple extension of ordinary triangle/quad subdivision.

31. Thank you !