Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and...

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Jorg Peters, SurfLab

Generalized spline subdivision

• Polynomial Heritage• Computing Moments• Shape and Eigenvalues

Jorg Peters SurfLab (Purdue,UFL)

Polynomial heritageof generalized spline subdivision

• Doo-Sabin

Catmull-Clark

• Increasing regions are regular: points and faces have standard valence

• Doo-Sabin bi-2 B-spline

• Catmull-Clark bi-3 B-spline

• Midedge Zwart-Powell C^1 box-spline

• Loop C^2 box-spline

box-spline = generalization of B-spline to shift-invariant partitions book: [de Boor, Hollig, Riemenschneider 94]

• Subdivision of the Zwart-Powell C^1 quadratic box-spline

c dSubdivision

Subdivision

basis function

Subdivision Rule

Subdivision

2 steps 4 steps

Zwart-Powell subdivision = 2 steps of Midedge subdivision

regular: 4-valence, quadrilaterals

Mid-edge Rule (“simplest rule”)

• Increasing regions are regular (polynomial)

• Union of surface layers at an extra-ordinary point

• Uses:

Representation as Bezier patches

Evaluation at non-binary points

Fast moment computation

Generalized spline subdivision

• Polynomial Heritage• Computing Moments• Shape and Eigenvalues

Jorg Peters SurfLab (Purdue,UFL)

Moments of objects enclosed by generalized subdivision surfaces

• Challenge: Exponential increase in the number of facets!

Volume

Inertia Frame

Center of mass

dUn f dS nn/ f f dV

dSnn/ f

Theory: Gauss’ Divergence Theorem:

The integral of the divergence over the volume

equals the integral of the normal component over the surface S

Theory: Change of variables

The area of the surface element S

equals the integral of the Jacobian |n| of the surface parametrization (x,y,z) over the domain U

dUn f dS nn/ f f dV

uvvu dvdu ]yx-y[x z dV1

For example, f=[0,0,z] n =

f n = z is piecewise polynomial in regular regions

Volume =

uvvu yxyx

ppatch Up

p dvdu )yxyx(z

“Volume” patch p = Up

p dvdu )yxyx(z

Schema for bi-3 Bezier patch

Volume patch p = Up

p dvdu )yxyx(z

Volume patch p = Up

p dvdu )yxyx(z

Volume patch p = Up

p dvdu )yxyx(z

Work: at each subdivision step linear for each extraordinary point add volume contribution of 3n patches

Doo-Sabin

p dvdu )yxyx(z

WV 0V iVi+1

iVlayer in p

Volume

• Error estimation: bounding boxes

• Geometric decay of error volume 1, 1/8, 1/64, ...

• Computing geometry given a fixed volume

Bisection

• Higher moments and the inertia frame

center of mass:

dV zdV,y ,dVx

inertia tensor:

dV,...xy ..., eigenvector frame

• Higher moments and the inertia frame

center of mass

• Physics-based animation

Center of mass support

• Simple registration, comparison

matching frames = computing a 3x3 matrix Q:

IP Q = IS

• Solution: Moments efficiently and exactly computed via Gauss’ theorem and polynomial heritage

Volume

Inertia Frame

Center of mass

Shape and eigenvalues

• Union of surface layers at an extra-ordinary point

• Control points transformed by the subdivision matrix

eigenvector expansion

• If all < 1, then collapse• If some > 1, then unbounded growth• Good sequence: 1, , , … where | | < 1• Eigenvectors of determine the tangent plane

Shape and eigenvalues• Fast contraction of 3-sided facets

= (1+cos(2pi/ 3))/2 = .25

• Slow contraction of large facets

= (1+cos(2pi/16))/2 = .962...midedge subdivision

• adjust subdominant eigenvalues (modified midedge subdivision)

<=> =. 5

Generalized spline subdivisionSummary

• Polynomial Heritage regular regions

• Computing MomentsGauss’ theorem

• Shape and Eigenvaluessubdominant values

Jorg Peters, SurfLab Generalized spline subdivision Polynomial Heritage Computing Moments Shape and...

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