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ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Extending Catmull-Clark Subdivision and
PCCM with Polar Structures
Ashish Myles Kestutis Karčiauskas Jörg Peters
Pacific GraphicsNovember 1, 2007
1 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Overview
2 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Overview
2 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Overview
2 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Overview
Valence 6= 4
2 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Overview
Catmull-Clarksubdivision
Valence 6= 4
2 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Overview
Catmull-Clarksubdivision
Valence 6= 4 PCCM
2 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Overview
Polarlayout
Catmull-Clarksubdivision
Valence 6= 4 PCCM
2 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Overview
New Bi-cubicpolar subdivision
Polarlayout
Catmull-Clarksubdivision
Valence 6= 4 PCCM
2 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Overview
New Bi-cubicpolar subdivision
Polarlayout
NewNURBS capping
Catmull-Clarksubdivision
Valence 6= 4 PCCM
2 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar structures appear naturally
Eye courtesy of ”Blender: Noob to Pro”
3 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Remove those unsightly wrinkles
Catmull-Clark Our method
4 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Make predictable ripples
Catmull-Clark Our method
5 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Model designers face the following challenges
Conventional picture
1. Align control mesh along features.
2. Use only quads.
3. Keep valence low.
6 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Model designers face the following challenges
New and improved picture
1. Align control mesh along features.
2. Use quads and polar structures.
3. Keep quad-mesh valence low. High polar valence OK!
6 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Model a face using polar structures and
multi-sided
7 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Model a face using polar structures and
multi-sided
7 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Model a face using polar structures and
multi-sided
7 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Model a face using polar structures and
multi-sided
7 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Model a face using polar structures and
multi-sided
7 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Model a face using polar structures and
multi-sided
7 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Model a face using polar structures and
multi-sided
7 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Model a face using polar structures and
multi-sided
7 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Model a face using polar structures and
multi-sided
7 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Modeling with polar connectivity
1. keeps the Catmull-Clark valence low,
2. shifts high-valence connectivity to polar structures, and
3. orients the control lines along model features(e.g. mouth).
8 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Mesh Refinement
Bi-cubicsubdivision
Polarlayout
NURBScapping
Catmull-Clarksubdivision
Valence 6= 4 PCCM
9 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar refinement
10 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar refinement
radial
radial
α
n
α
n
α
n
1−α
1−β
γi+1
γi−1
γi
18
68
18
12
12
α :=β −1
4, β :=
5
8,
ck
n:= cos
„
2πk
n
«
,
γk :=1
n
„
β −1
2+
5
8ck
n
+(ck
n)2
+1
2(c
k
n)3
«
10 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar refinement
radial circular
radial circular
α
n
α
n
α
n
1−α
1−β
γi+1
γi−1
γi
18
68
18
12
12
18 6
818
12
12
α :=β −1
4, β :=
5
8,
ck
n:= cos
„
2πk
n
«
,
γk :=1
n
„
β −1
2+
5
8ck
n
+(ck
n)2
+1
2(c
k
n)3
«
10 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar is easily combined with Catmull-Clark
11 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar is easily combined with Catmull-Clark
11 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar is easily combined with Catmull-Clark
11 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar is easily combined with Catmull-Clark
11 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar is easily combined with Catmull-Clark
11 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar is easily combined with Catmull-Clark
11 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar is easily combined with Catmull-Clark
11 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar is easily combined with Catmull-Clark
⇒ C 1 with bounded curvature at the polar limit point.
◮ Verified using standard analysis tools from subdivisiontheory.
11 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Results
12 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
NURBS Constructions
Bi-cubicsubdivision
Polarlayout
NURBScapping
Catmull-Clarksubdivision
Valence 6= 4 PCCM
13 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar structures can be C 1 capped by a single
NURBS patch
original control mesh
14 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar structures can be C 1 capped by a single
NURBS patch
original control mesh
k-times subdivided mesh
1. subdivide
radially
14 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Polar structures can be C 1 capped by a single
NURBS patch
original control mesh
k-times subdivided mesh periodic B-spline
1. subdivide
radially
2. project
14 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Capping Polar with a single NURBS patch
15 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Capping Polar with a single NURBS patch
15 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Capping Polar with a single NURBS patch
15 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Capping Polar with a single NURBS patch
15 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Capping Polar with a single NURBS patch
15 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Capping Polar with a single NURBS patch
15 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Capping Polar with a single NURBS patch
15 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Capping Polar with a single NURBS patch
⇒ C 1 with bounded curvature at the polar limit point.
◮ Singular parametrizations typically tricky◮ Our B-spline patch = limit surface of a particular
subdivision scheme◮ Analyze using subdivision machinery!
15 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Results
16 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Conclusion
Bi-cubicsubdivision
Polarlayout
NURBScapping
Catmull-Clarksubdivision
Valence 6= 4 PCCM
17 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Questions?
Bi-cubicsubdivision
Polarlayout
NURBScapping
Catmull-Clarksubdivision
Valence 6= 4 PCCM
17 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Backup Slides
18 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
High-valent Catmull-Clark layout → polar layout
19 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
High-valent Catmull-Clark layout → polar layout
19 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Analysis
...
...
c
Ac
A
0i0i
1010
1111
1212
20
2021
21
30
30
31
31
A =
A0 A1 ... An−1An−1 A0 ... An−2...
. . ....
A1 ... An−1 A0
A0 :=
1/n 0 0 01/n Γ0 0 00 3/4 1/4 00 3/16 11/16 1/8
, Ai :=
[1/n 0 0 01/n Γi 0 00 0 0 00 0 0 0
]
.
20 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Analysis
Spectral analysis of A (circulant) gives desired properties:
◮ Eigenvalues: 1,
1st & n − 1st
Fourier blocks︷ ︸︸ ︷
1/2, 1/2 ,
n︷ ︸︸ ︷
1/4, . . . , 1/4.
◮ Characteristic map is regular (⇒ C 1).
◮ Geometric multiplicities = algebraic multiplicities for A.
⇒ Bounded curvature.
21 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Tensor product B-spline refinement
Polar
Catmull-Clark
22 / 23
ExtendingCatmull-Clark
Subdivision andPCCM with Polar
Structures
A. MylesK. Karčiauskas
J. Peters
Polar Structure
Examples
Modeling
Refinement
Polar refinement
General meshrefinement
NURBSConstructions
Separating Catmull-Clark and polar extraordinary
limit points
radial circular
α
nα
nα
n
1−α
1−β
γi+1
γi−1
γi
18
68
18
12
12
18 6
818
12
12
23 / 23
Polar StructureExamplesModeling
RefinementPolar refinementGeneral mesh refinement
NURBS Constructions