1. On Triangle/Quad Subdivision Scott Schaefer and Joe Warren TOG 22(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

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. 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 Previous works

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

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: 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: Extraordinary vertex (not valence four vertex) After one iteration

12. Factorization Step2. Weighted averagingStep1. Linear subdivision Averaging mask for regular vertex

13. Centroid averaging approach (a) Computation of centroids (b) Averaging the 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 Surface C. T. Loop, 1987 Original meshApplying subdivision once Extraordinary vertex (not valence six vertex)

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 Subdivision Stam, J. and Loop, C., 2003 1. Initial shape2. Linear subdivision3. 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 I. “ Unzips ” the mesh into disjoint pieces consisting of only triangles or only quads. (Levin and Levin [2003]) II. Linear subdivision. (Stam and Loop [2003]) III. Weighted average of centroids. (Warren and Schaefer [2004]) The unified subdivision scheme

28. Unzipping pass 1. Identify edges on the surface contained by both triangles and quads. 2. Apply the unzipping masks (, ) to this curve network.unzipping masks 3. Linear subdivision. 4. Weighted average of centroids

29. Unzipping mask Back Unzipping masks for the vertices part of the triangle/quad boundary in the ordinary case (top) and the arbitrary case (down)

30. 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.

31. 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.

32. Thank you !