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

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

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### On Triangle/Quad SubdivisionScott Schaefer and Joe WarrenTOG22(1) 28–36, 2005

Reporter: Chen zhonggui

2005.10.27

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.

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.

Outline

- Preview
- Previous works
- Catmull-Clark surface
- Loop surface
- Triangle/Quad Subdivision
- On triangle/Quad Subdivision
- Conclusion

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

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

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

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

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)

Property

- continuous on the regular quad regions.
- continuous at extraordinary vertices.

Loop SurfaceC. T. Loop, 1987

Extraordinary vertex

(not valence six vertex)

Original mesh

Applying subdivision once

Property

- continuous on the regular triangle regions.
- continuous at extraordinary vertices but valence three vertices (valence three vertices are only ).

Demo

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.

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.

Triangle/Quad SubdivisionStam, J. and Loop, C., 2003

1. Initial shape

2. Linear subdivision

3. Weighted averaging

Averaging masks

(a) Averaging masks for ordinary quad-triangles

(b) Averaging mask for extraordinary vertex?

Weighted centroid averaging approach

(a) Centroids are weighted by their angular contribution

(b) The result averaging masks

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

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])

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

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.

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.

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