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
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