Subdivision Surfaces - Connelly Barnes€¦ · Subdivision Surfaces Connelly Barnes CS 4810: Graphics Acknowledgment: slides by Jason Lawrence, Misha Kazhdan, Allison Klein, Tom Funkhouser,

Subdivision Surfaces

Connelly Barnes

CS 4810: Graphics

Acknowledgment: slides by Jason Lawrence, Misha Kazhdan, Allison Klein, Tom Funkhouser, Adam Finkelstein and David Dobkin

Subdivision• How do you make a smooth curve?

Zorin & Schroeder SIGGRAPH 99 Course Notes

We want to “smooth out” severe angles







Subdivision Surfaces• Coarse mesh & subdivision ruleoDefine smooth surface as limit of

sequence of refinements


Key Questions• How to subdivide the mesh?oAim for properties like smoothness

• How to store the mesh?oAim for efficiency of implementing subdivision rules


General Subdivision Scheme• How to subdivide the mesh?

Two parts:»Refinement:

–Add new vertices and connect (topological)»Smoothing:

–Move vertex positions (geometric)

Loop Subdivision Scheme• How to subdivide the mesh?

Refinement:»Subdivide each triangle into 4 triangles by splitting each edge and connecting new vertices


Loop Subdivision Scheme


Existing vertex being moved from one level to the next

• How to subdivide the mesh:RefinementSmoothing: »Existing Vertices: Choose new location as weighted average of original vertex and its neighbors

Loop Subdivision Scheme• General rule for moving existing interior vertices:


What about vertices that have more Or less than 6 neighboring faces?

new_position = (1-kβ)original_position + sum(β*each_original_vertex)

What about vertices that have more or less than 6 neighboring faces?

Loop Subdivision Scheme• General rule for moving existing interior vertices:


What about vertices that have more Or less than 6 neighboring faces?

new_position = (1-kβ)original_position + sum(β*each_original_vertex)

What about vertices that have more or less than 6 neighboring faces?

0≤ β ≤1/k:

• As β increases, the contribution from adjacent vertices plays a more important role.

Where do existing vertices move?• How to choose β?oAnalyze properties of limit surfaceoInterested in continuity of surface and smoothnessoInvolves calculating eigenvalues of matrices

»Original Loop

»Warren

Loop Subdivision Scheme• How to subdivide the mesh:

RefinementSmoothing:»Inserted Vertices: Choose location as weighted average of original vertices in local neighborhood

New vertex being inserted


Boundary Cases?• What about extraordinary vertices and boundary

edges?:oExisting vertex adjacent to a missing triangleoNew vertex bordered by only one triangle


Boundary Cases?• Rules for extraordinary vertices and boundaries:


1/2 1/2 1/8 1/83/4


Pixar


PixarZorin & Schroeder SIGGRAPH 99 Course Notes


Geri’s Game, Pixar

Subdivision Schemes• There are different subdivision schemesoDifferent methods for refining topology oDifferent rules for positioning vertices

»Interpolating versus approximating

Zorin & Schroeder, SIGGRAPH 99 , Course Notes

Subdivision Schemes


21

Key Questions• How to refine the mesh?oAim for properties like smoothness

• How to store the mesh?oAim for efficiency for implementing subdivision rules


Subdivision SmoothnessTo determine the smoothness of the subdivision:

• Repeatedly apply the subdivision scheme

• Look at the neighborhood in the limit.































…

Computing infinitely many iterations is computationally prohibitive!!!

Subdivision Matrix• Compute the new positions/vertices as a linear

combination of previous ones.

p0

p1p6

p5

p4 p3

p2 p0’p5’

p4’ p3’

p2’

p1’p6’



p0

p1p6

p5

p4 p3

p2 p0’p5’

p4’ p3’

p2’

p1’p6’

Subdivision Matrix0

BBBBBBBB@

p00

p01

p02

p03

p04

p05

p06

1

CCCCCCCCA

=116

0

BBBBBBBB@

10 1 1 1 1 1 16 6 2 0 0 0 26 2 6 2 0 0 06 0 2 6 2 0 06 0 0 2 6 2 06 0 0 0 2 6 26 2 0 0 2 2 6

1

CCCCCCCCA

0

BBBBBBBB@

p0

p1

p2

p3

p4

p5

p6

1

CCCCCCCCA



• To find the limit position of p0, repeatedly apply the subdivision matrix.

0

BBBBBBBBBB@

p(n)0

p(n)1

p(n)2

p(n)3

p(n)4

p(n)5

p(n)6

1

CCCCCCCCCCA

=116

0

BBBBBBBB@

10 1 1 1 1 1 16 6 2 0 0 0 26 2 6 2 0 0 06 0 2 6 2 0 06 0 0 2 6 2 06 0 0 0 2 6 26 2 0 0 2 2 6

1

CCCCCCCCA

n 0

BBBBBBBB@

p0

p1

p2

p3

p4

p5

p6

1

CCCCCCCCA



• To find the limit position of p0, repeatedly apply the subdivision matrix.

0

BBBBBBBBBB@

p(n)0

p(n)1

p(n)2

p(n)3

p(n)4

p(n)5

p(n)6

1

CCCCCCCCCCA

=116

0

BBBBBBBB@

10 1 1 1 1 1 16 6 2 0 0 0 26 2 6 2 0 0 06 0 2 6 2 0 06 0 0 2 6 2 06 0 0 0 2 6 26 2 0 0 2 2 6

1

CCCCCCCCA

n 0

BBBBBBBB@

p0

p1

p2

p3

p4

p5

p6

1

CCCCCCCCA

If, after a change of basis we have M=A-1DA, where D is a diagonal matrix, then:

Mn=A-1DnA,

Since D is diagonal, raising D to the n-th power just amounts to raising each of the diagonal entries of D to the n-th power.

Subdivision Modeling• ZBrush Modeling Session

38

https://www.youtube.com/watch?v=iz4Gf-qlNbM

Key Questions• How to refine the mesh?oAim for properties like smoothness

• How to store the mesh?oAim for efficiency for implementing subdivision rules


Polygon Meshes• Mesh RepresentationsoIndependent facesoVertex and face tablesoAdjacency listsoWinged-Edge

Independent Faces• Each face lists vertex coordinates

Independent Faces• Each face lists vertex coordinates

ûRedundant verticesûNo topology information

Vertex and Face Tables• Each face lists vertex references


üShared vertices


üShared verticesûStill no topology information

Adjacency Lists• Store all vertex, edge, and face adjacencies


üEfficient topology traversal


üEfficient topology traversalûExtra storageûVariable size arrays

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Subdivision Surfaces - Connelly Barnes€¦ · Subdivision Surfaces Connelly Barnes CS 4810: Graphics Acknowledgment: slides by Jason Lawrence, Misha Kazhdan, Allison Klein, Tom Funkhouser,