Extending Catmull-Clark Subdivision and PCCM with Polar … · Catmull-Clark Subdivision and PCCM...

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

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Overview

2 / 23



Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Overview

Valence 6= 4

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Overview

Catmull-Clarksubdivision

Valence 6= 4

2 / 23



Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Overview


Valence 6= 4 PCCM

2 / 23



Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Overview

Polarlayout


Valence 6= 4 PCCM

2 / 23



Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Overview

New Bi-cubicpolar subdivision

Polarlayout


Valence 6= 4 PCCM

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Overview

New Bi-cubicpolar subdivision

Polarlayout

NewNURBS capping


Valence 6= 4 PCCM

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Polar structures appear naturally

Eye courtesy of ”Blender: Noob to Pro”

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Remove those unsightly wrinkles

Catmull-Clark Our method

4 / 23



Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Make predictable ripples

Catmull-Clark Our method

5 / 23



Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Model designers face the following challenges

Conventional picture

1. Align control mesh along features.

2. Use only quads.

3. Keep valence low.

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


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!

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Model a face using polar structures and

multi-sided

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


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).

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Mesh Refinement

Bi-cubicsubdivision

Polarlayout

NURBScapping


Valence 6= 4 PCCM

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Polar refinement

10 / 23



Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


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



Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


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



Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Polar is easily combined with Catmull-Clark

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


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.

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Results

12 / 23



Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

NURBS Constructions

Bi-cubicsubdivision

Polarlayout

NURBScapping


Valence 6= 4 PCCM

13 / 23



Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Polar structures can be C 1 capped by a single

NURBS patch

original control mesh

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions


NURBS patch


k-times subdivided mesh

1. subdivide

radially

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions


NURBS patch


k-times subdivided mesh periodic B-spline

1. subdivide

radially

2. project

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Capping Polar with a single NURBS patch

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


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!

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Results

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Conclusion

Bi-cubicsubdivision

Polarlayout

NURBScapping


Valence 6= 4 PCCM

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Questions?

Bi-cubicsubdivision

Polarlayout

NURBScapping


Valence 6= 4 PCCM

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Backup Slides

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

High-valent Catmull-Clark layout → polar layout

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


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

]

.

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


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.

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


NURBSConstructions

Tensor product B-spline refinement

Polar

Catmull-Clark

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Structures


J. Peters

Polar Structure

Examples

Modeling

Refinement

Polar refinement


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

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Extending Catmull-Clark Subdivision and PCCM with Polar … · Catmull-Clark Subdivision and PCCM...

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