subdivision curves and surfacessheffa/dgp/ppts/subdivision.pdf · subdivision surfaces or how to...

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University ofUniversity ofBritish ColumbiaBritish Columbia

Subdivision Surfaces or How to Generate a Smooth Mesh?

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Subdivision Curves and Surfaces

Subdivision – given polyline(2D)/mesh(3D) recursively modify & add vertices to achieve smooth curve/surface

Each iteration generates smoother + more refined mesh

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What is wrong with NURBS?

Very, very difficult to avoid seamsWoody’s hand from Pixar’s Toy Story

Subdivision – no seamsGeri’s hand from Geri’s Game


Corner Cutting

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Corner Cutting

1 : 33

: 1


Corner Cutting

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Corner Cutting


Corner Cutting

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Corner Cutting


Corner Cutting

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Corner Cutting


Corner Cutting

control polygon

limit curvecontrol point

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The 4-point scheme


The 4-point scheme

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The 4-point scheme

1 :

1

1 :

1


The 4-point scheme

1 :

8

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The 4-point scheme


The 4-point scheme

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The 4-point scheme


The 4-point scheme

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The 4-point scheme


The 4-point scheme

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The 4-point scheme


The 4-point scheme

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The 4-point scheme


The 4-point scheme

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The 4-point scheme


The 4-point scheme

control polygon

limit curvecontrol point

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Subdivision curves

Non interpolatory subdivision schemes

Corner Cutting

Interpolatory subdivision schemes

The 4-point scheme


Basic concepts of Subdivision

Subdivision curve – limit of recursive subdivision applied to given polygonEach iteration

Increase number of vertices (approximately) * 2Initial polygon - control polygon

Central questions: Convergence: Given a subdivision operator and a control polygon, does the subdivision process converge? Smoothness: Does subdivision converge to smooth curve?

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Subdivision schemes for surfaces

Each iteration Subdivision refines control net (mesh)Increase number of vertices (approximately) * 4

Mesh vertices converge to limit surfaceEvery subdivision method has:

Method to generate net topology rules to calculate location of new vertices


Defined for triangular meshes (control nets)

Every face replaced by 4 new triangular facesTwo kinds of new vertices:

Green vertices are associated with old edgesYellow vertices are associated with old vertices

Triangular subdivision

Old vertices New vertices

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Loop’s scheme

New vertex is weighted average of old vertices

List of weights called subdivision mask or stencil

3 3

1

1

1

1

1

1

1

( )( )n

nnwn −

+−= 22cos2340

64π

Rule for new yellow vertices (n – vertex valence)

Rule for new green vertices

wn


The original control net

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After 1st iteration


After 2nd iteration

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After 3rd iteration


Loop Scheme

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Loop Limit Surface

Limit surfaces of Loop’s subdivision is C2 almost everywhere


Butterfly Scheme

Interpolatory scheme

New yellow vertices inherit location of old vertices

New green vertices calculated by following stencil:-1

-1

-1

-1

88

2

2

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The original control net


After 1st iteration

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After 2nd iteration


After 3rd iteration

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Butterfly Scheme


Butterfly limit surface

Limit surfaces of Butterfly subdivision are C1, but do not have second derivative

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Comparison


Boundaries & Features: Loop

Boundary vertices – depend ONLY on other boundary vertices Corner vertices left in place

Sometimes modified rules for corner neighbors

Treat features (creases) like boundaries

1 6 1

1 1

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Scheme Zoo

More schemes:

Catmul-Clark KobbeltDuo-Sabin…

Proving scheme works:ConvergenceDegree of continuityAffine invariance


Affine Invariance

TransformT

TransformT

subdivide

subdivide

Coefficients of masks sum to 1 –weighted average

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Affine Invariance

1

8

1

8

3

83

8

®

® ®

® ®

1¡ n® p =X

aipi

Coefficients of masks must sum to 1

Xai(pi + t) =

³Xai´t+ p

1

displacement


Analysis of Subdivision

Test:ConvergenceSmoothness

Goals:Help to choose the rules Ensure that all surfaces have desired properties

PlanDefine subdivision surfacesRelate properties to coefficients

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Subdivision Matrix

Relate control on finer level to coarser level

Useful forAnalysis of properties

smoothnessaffine invariance

Formulas for normals

Explicit evaluation of surfaces at arbitrary points of the domain


Controls of K-Sided Patch

Simplest case - consider only 1-ring

c p0p6p4

p5

p3p2

c

p6

p0

p1p2

p3

p5

p4

p1

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0BBBBBBBBBBBBBBBBBBBBBBBBB@

7=16 3=16 3=16 3=16 0 0 0 0 0 0

3=8 3=8 1=8 1=8 0 0 0 0 0 0

3=8 1=8 3=8 1=8 0 0 0 0 0 0

3=8 1=8 1=8 3=8 0 0 0 0 0 0

1=16 5=8 1=16 1=16 1=16 0 0 1=16 0 1=16

1=16 1=16 5=8 1=16 0 1=16 0 1=16 1=16 0

1=16 1=16 1=16 5=8 0 0 1=16 0 1=16 1=16

1=8 3=8 3=8 0 0 0 0 1=8 0 0

1=8 0 3=8 3=8 0 0 0 0 1=8 0

1=8 3=8 0 3=8 0 0 0 0 0 1=8

1CCCCCCCCCCCCCCCCCCCCCCCCCA

Subdivision Matrix



7=16 3=16 3=16 3=16 0 0 0 0 0 0

3=8 3=8 1=8 1=8 0 0 0 0 0 0

3=8 1=8 3=8 1=8 0 0 0 0 0 0

3=8 1=8 1=8 3=8 0 0 0 0 0 0

1=16 5=8 1=16 1=16 1=16 0 0 1=16 0 1=16

1=16 1=16 5=8 1=16 0 1=16 0 1=16 1=16 0

1=16 1=16 1=16 5=8 0 0 1=16 0 1=16 1=16

1=8 3=8 3=8 0 0 0 0 1=8 0 0

1=8 0 3=8 3=8 0 0 0 0 1=8 0

1=8 3=8 0 3=8 0 0 0 0 0 1=8


Subdivision Matrix

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7=16 3=16 3=16 3=16 0 0 0 0 0 0

3=8 3=8 1=8 1=8 0 0 0 0 0 0

3=8 1=8 3=8 1=8 0 0 0 0 0 0

3=8 1=8 1=8 3=8 0 0 0 0 0 0

1=16 5=8 1=16 1=16 1=16 0 0 1=16 0 1=16

1=16 1=16 5=8 1=16 0 1=16 0 1=16 1=16 0

1=16 1=16 1=16 5=8 0 0 1=16 0 1=16 1=16

1=8 3=8 3=8 0 0 0 0 1=8 0 0

1=8 0 3=8 3=8 0 0 0 0 1=8 0

1=8 3=8 0 3=8 0 0 0 0 0 1=8


Subdivision Matrix



7=16 3=16 3=16 3=16 0 0 0 0 0 0

3=8 3=8 1=8 1=8 0 0 0 0 0 0

3=8 1=8 3=8 1=8 0 0 0 0 0 0

3=8 1=8 1=8 3=8 0 0 0 0 0 0

1=16 5=8 1=16 1=16 1=16 0 0 1=16 0 1=16

1=16 1=16 5=8 1=16 0 1=16 0 1=16 1=16 0

1=16 1=16 1=16 5=8 0 0 1=16 0 1=16 1=16

1=8 3=8 3=8 0 0 0 0 1=8 0 0

1=8 0 3=8 3=8 0 0 0 0 1=8 0

1=8 3=8 0 3=8 0 0 0 0 0 1=8


Subdivision Matrix

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7=16 3=16 3=16 3=16 0 0 0 0 0 0

3=8 3=8 1=8 1=8 0 0 0 0 0 0

3=8 1=8 3=8 1=8 0 0 0 0 0 0

3=8 1=8 1=8 3=8 0 0 0 0 0 0

1=16 5=8 1=16 1=16 1=16 0 0 1=16 0 1=16

1=16 1=16 5=8 1=16 0 1=16 0 1=16 1=16 0

1=16 1=16 1=16 5=8 0 0 1=16 0 1=16 1=16

1=8 3=8 3=8 0 0 0 0 1=8 0 0

1=8 0 3=8 3=8 0 0 0 0 1=8 0

1=8 3=8 0 3=8 0 0 0 0 0 1=8


Subdivision Matrix



7=16 3=16 3=16 3=16 0 0 0 0 0 0

3=8 3=8 1=8 1=8 0 0 0 0 0 0

3=8 1=8 3=8 1=8 0 0 0 0 0 0

3=8 1=8 1=8 3=8 0 0 0 0 0 0

1=16 5=8 1=16 1=16 1=16 0 0 1=16 0 1=16

1=16 1=16 5=8 1=16 0 1=16 0 1=16 1=16 0

1=16 1=16 1=16 5=8 0 0 1=16 0 1=16 1=16

1=8 3=8 3=8 0 0 0 0 1=8 0 0

1=8 0 3=8 3=8 0 0 0 0 1=8 0

1=8 3=8 0 3=8 0 0 0 0 0 1=8


Subdivision Matrix

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7=16 3=16 3=16 3=16 0 0 0 0 0 0

3=8 3=8 1=8 1=8 0 0 0 0 0 0

3=8 1=8 3=8 1=8 0 0 0 0 0 0

3=8 1=8 1=8 3=8 0 0 0 0 0 0

1=16 5=8 1=16 1=16 1=16 0 0 1=16 0 1=16

1=16 1=16 5=8 1=16 0 1=16 0 1=16 1=16 0

1=16 1=16 1=16 5=8 0 0 1=16 0 1=16 1=16

1=8 3=8 3=8 0 0 0 0 1=8 0 0

1=8 0 3=8 3=8 0 0 0 0 1=8 0

1=8 3=8 0 3=8 0 0 0 0 0 1=8


Subdivision Matrix



7=16 3=16 3=16 3=16 0 0 0 0 0 0

3=8 3=8 1=8 1=8 0 0 0 0 0 0

3=8 1=8 3=8 1=8 0 0 0 0 0 0

3=8 1=8 1=8 3=8 0 0 0 0 0 0

1=16 5=8 1=16 1=16 1=16 0 0 1=16 0 1=16

1=16 1=16 5=8 1=16 0 1=16 0 1=16 1=16 0

1=16 1=16 1=16 5=8 0 0 1=16 0 1=16 1=16

1=8 3=8 3=8 0 0 0 0 1=8 0 0

1=8 0 3=8 3=8 0 0 0 0 1=8 0

1=8 3=8 0 3=8 0 0 0 0 0 1=8


Subdivision Matrix

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7=16 3=16 3=16 3=16 0 0 0 0 0 0

3=8 3=8 1=8 1=8 0 0 0 0 0 0

3=8 1=8 3=8 1=8 0 0 0 0 0 0

3=8 1=8 1=8 3=8 0 0 0 0 0 0

1=16 5=8 1=16 1=16 1=16 0 0 1=16 0 1=16

1=16 1=16 5=8 1=16 0 1=16 0 1=16 1=16 0

1=16 1=16 1=16 5=8 0 0 1=16 0 1=16 1=16

1=8 3=8 3=8 0 0 0 0 1=8 0 0

1=8 0 3=8 3=8 0 0 0 0 1=8 0

1=8 3=8 0 3=8 0 0 0 0 0 1=8


Subdivision Matrix



7=16 3=16 3=16 3=16 0 0 0 0 0 0

3=8 3=8 1=8 1=8 0 0 0 0 0 0

3=8 1=8 3=8 1=8 0 0 0 0 0 0

3=8 1=8 1=8 3=8 0 0 0 0 0 0

1=16 5=8 1=16 1=16 1=16 0 0 1=16 0 1=16

1=16 1=16 5=8 1=16 0 1=16 0 1=16 1=16 0

1=16 1=16 1=16 5=8 0 0 1=16 0 1=16 1=16

1=8 3=8 3=8 0 0 0 0 1=8 0 0

1=8 0 3=8 3=8 0 0 0 0 1=8 0

1=8 3=8 0 3=8 0 0 0 0 0 1=8


Subdivision Matrix

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Diagonalize subdivision matrix

eigenvectors

eigenvalues

: vector of points in a neighborhood

xi, i = 0::N

¸i, i = 0::N

p

p =Xi

aixi

(N+1)-vector of 3D points3D vector

Eigen Decomposition


“Good” case: &

can make zero by moving control points (by affine invariance)

a0

Smp = a0x0 + ¸m1 a1x1 + ¸

m2 a2x2 + ¸

m3 a3x3 + : : :

x0 = [1;1; : : : 1]

Eigenvectors

limit position tangent vectors

10 =λ 1n,,1i,1i −=< Kλ

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Next higher order termsassume move control points so that

1

¸mSmp = (a1x1 + a2x2) +

µ¸3¸

¶mx3 + : : :

vanishes

a0 = 0¸ = ¸1 = ¸2 > j¸3j

subdominant eigenvectors

Subdominant Eigenvectors


Loop schemeOnly one ring is necessary – matrix of K+1 rows

Compute eigenvectors (left l & right x)

Get coefficients from projection

Example:Tangents

),( Pla ii =

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c p0p6p4

p5

p3p2

c

p6

p0

p1p2

p3

p5

p4

p1

Eigenvalue 0left and right vector are different

lcenter0 = d

d=3

3+ 8k®

li0 =8

3d®; i = 0 : : :K ¡ 1

Example: Limit Position

a0 = dc+8

3d®Xi

pi


Subdivision Drawbacks

Not always intuitiveCan have artifactsHard to control

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Properties

Works best on regular connectivity (valence 6)Easy to implement

Efficiency is another matterUse Eigen-analysis for exact computation

Local support – operations depend on local data Order does not matterData structure support

Allow LOD View mesh at any level as control mesh

Apply geometric modifications

ContinuousScheme dependent


Geri’s Game

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