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

Factorization

Step2. Weighted averagingStep1. Linear subdivision

Averaging mask for regular vertex

Centroid averaging approach

(a) Computation of centroids

(b) Averaging the centroids

Subdivision Matrix(1)

1 1

(1)

(1)

nn

Q Q

M

QQ

VV

V

1Q 3Q

nQ

One-ring neighboring vertices of extraordinary vertex V

( )1 1

( )

( )

k

k

knn

k

Q Q

M

QQ

VV

(1)1Q

M: a constant matrix

2Q

Property continuous on the regular quad regi

ons. continuous at extraordinary vertices.

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Loop SurfaceC. T. Loop, 1987

Original mesh Applying subdivision once

Extraordinary vertex(not valence six vertex)

Loop Surface

(1) Averaging mask for regular vertex

(2) Averaging mask for extraordinary vertex ?

Centroid averaging approach

(a) Centroid calculation for triangles

(b) The result averaging mask

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

Averaging mask for regular quads

Averaging mask for regular triangles

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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Thank you !