Scott SchaeferScott SchaeferJoe WarrenJoe Warren

A Factored, Interpolatory Subdivision for Surfaces

of Revolution

Rice University

Allows coarse, low-polygon models to Allows coarse, low-polygon models to approximate smooth shapesapproximate smooth shapes

Importance of Subdivision

Subdivision A process that takes a polygon as input A process that takes a polygon as input

and produces a new polygon as outputand produces a new polygon as output

Defines a sequence which should converge in Defines a sequence which should converge in the limitthe limit

1kP

kP

11 kkk PSP

Interpolatory Subdivision Subdivision scheme is Subdivision scheme is interpolatoryinterpolatory if the if the

vertices of are a subset of the vertices vertices of are a subset of the vertices ofof

Example: linear subdivisionExample: linear subdivision

1kP

kP

Interpolatory Scheme Place new point on curve defined by a cubic Place new point on curve defined by a cubic

interpolant through 4 consecutive pointsinterpolant through 4 consecutive points[Deslauriers and Dubuc, 1989][Deslauriers and Dubuc, 1989]

If parameterization is uniform, weights do not depend If parameterization is uniform, weights do not depend on scaleon scale

1C

0.5 1 1.5 2 2.5 3t

0.5

1

1.52

2.5

33.5

4f (t)

9 16___

9 16___

-1 16___

-1 16___

Curve Subdivision Example

Produces a curve that isProduces a curve that is Cannot reproduce circlesCannot reproduce circles

1C

Extension to Surfaces Extended to quadrilateral surfaces of arbitrary Extended to quadrilateral surfaces of arbitrary

topology [Kobbelt, 1995]topology [Kobbelt, 1995] Surface subdivision scheme is Surface subdivision scheme is

[Zorin, 2000][Zorin, 2000]

1C

Modeling Circles

2

2

2

11)(

12)(

ttty

tttx

ntty

nttx

2sin)(

2cos)(

An Interpolatory Scheme for Circles

Use a different set of interpolating functions Use a different set of interpolating functions to compute weights for new verticesto compute weights for new vertices

Solve for weights like beforeSolve for weights like before Capable of reproducing global functionsCapable of reproducing global functions

represent circlesrepresent circles

)sin(),cos(,,1 ttt

)sin(),cos( tt

Form of the Weights Weights depend on level of subdivision Weights depend on level of subdivision

Limit is of non-stationary scheme is Limit is of non-stationary scheme is [Dyn and Levin, 1995][Dyn and Levin, 1995]

21 1

nn

12

2

1

nn

nw

1C

0.5 1 1.5 2 2.5 3t

0.5

1

1.52

2.5

33.5

4f(t)

-w 16

n__

8+w 16

n___

-w 16

n__8+w 16

n___

Geometric Interpretation of Weights is a tension associated with subdivision schemeis a tension associated with subdivision scheme Tensions determine how much the curve pulls away Tensions determine how much the curve pulls away

from edges of original polygonfrom edges of original polygon

To produce a circle choose to beTo produce a circle choose to be0

n2cos

0

Factoring the Subdivision Step

Factor into linear subdivision followed by Factor into linear subdivision followed by differencing differencing

1kS

111 kkk LDS

The Differencing Mask

Linear subdivision isolates the addition of Linear subdivision isolates the addition of new verticesnew vertices

Differencing repositions verticesDifferencing repositions vertices Rule is uniformRule is uniform

Extension to Surfaces

Linear subdivision Bilinear subdivisionLinear subdivision Bilinear subdivision Differencing Two-dimensional differencingDifferencing Two-dimensional differencing Use tensor productUse tensor product

Surface Example

Linear subdivision + DifferencingLinear subdivision + Differencing Subdivision method for curve networksSubdivision method for curve networks

Example: Circular Torus

Tensions set to zero to produce a circleTensions set to zero to produce a circle

Cylinder Example

Open boundary converges to a circle as wellOpen boundary converges to a circle as well

Extensions

Open meshesOpen meshes Extraordinary verticesExtraordinary vertices Non-manifold geometryNon-manifold geometry Tagged meshes for creasesTagged meshes for creases

Demo

Construct profile curve to define surfaces of Construct profile curve to define surfaces of revolutionrevolution

Conclusions

Developed curve scheme to produce circlesDeveloped curve scheme to produce circles Tensions control shape of the curveTensions control shape of the curve Factored subdivision into linear subdivision Factored subdivision into linear subdivision

plus differencingplus differencing Extended to surfacesExtended to surfaces