On Triangle/Quad Subdivision Scott Schaefer and Joe Warren TOG 22(1) 28 – 36 , 2005
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Transcript of On Triangle/Quad Subdivision Scott Schaefer and Joe Warren TOG 22(1) 28 – 36 , 2005
On Triangle/Quad SubdivisionScott Schaefer and Joe Warren
TOG 22(1) 28–36, 2005
Reporter: Chen zhonggui 2005.10.27
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About the authors
Scott Schaefer: B.S in computer science and mathemat
ics, Trinity University M.S. in computer science, Rice University Ph.D. candidate at Rice University Research interests: computer graph
ics and computer-aided geometric design.
About the authors
Joe Warren: Professor of computer science at Rice U
niversity Associate editor of TOG B.S. in computer science, math, and ele
ctrical engineering, Rice University M.S. and Ph.D. in computer science, Co
rnell University Research interests: subdivision, geo
metric modeling, and visualization.
Outline Preview Previous works Catmull-Clark surface Loop surface Triangle/Quad Subdivision On triangle/Quad Subdivision Conclusion
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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 Triang
les.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
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
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Catmull-Clark SurfaceE. 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
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:
2 ( 3)Q R S n
n n n
Extraordinary vertex(not valence four verte
x)
After one iteration
Subdivision Matrix(1)
1 1
(1)
(1)
nn
Q Q
M
VV
V
1Q 3Q
nQ
One-ring neighboring vertices of extraordinary vertex V
( )1 1
( )
( )
k
k
knn
k
Q Q
M
VV
(1)1Q
M: a constant matrix
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Loop SurfaceC. T. Loop, 1987
Original mesh Applying subdivision once
Extraordinary vertex(not valence six vertex)
Property continuous on the regular triangle r
egions. continuous at extraordinary vertices
but valence three vertices (valence three vertices are only ).
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Demo
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Drawbacks of above surfaces Catmull-Clark surfaces behave very p
oorly on triangle-only base meshes:
A regular triangular mesh (left) behaves poorlywith 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 qua
d 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 qua
d and triangle regions. Cannot be along the quad/triangle bound
ary.
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Demo
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])
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The unified subdivision scheme
Unzipping pass
1. Identify edges on the surface contained by both triangles and quads.
2. Apply the unzipping masks ( , ) to this curve network.
3. Linear subdivision.4. Weighted average of centroids
tU qU
Property continuous on both the regular qua
d and the triangle regions of the mesh. continuous along the quad/triangle
boundary. continuous at the irregular quad an
d triangle regions.
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Conclusion We have presented a subdivision sche
me 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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