Scott Schaefer

1
A Factored, Interpolatory Subdivision for
Surfaces of Revolution

• Scott Schaefer
• Joe Warren

Rice University
2
Importance of Subdivision

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

3
Subdivision

• A process that takes a polygon as input and
  produces a new polygon as output
• Defines a sequence which should converge in the
  limit

4
Interpolatory Subdivision

• Subdivision scheme is interpolatory if the
  vertices of are a subset of the vertices
• of
• Example linear subdivision

5
Interpolatory Scheme

• Place new point on curve defined by a cubic
  interpolant through 4 consecutive points
• Deslauriers and Dubuc, 1989
• If parameterization is uniform, weights do not
  depend on scale

6
Curve Subdivision Example

• Produces a curve that is
• Cannot reproduce circles

7
Extension to Surfaces

• Extended to quadrilateral surfaces of arbitrary
  topology Kobbelt, 1995
• Surface subdivision scheme is
• Zorin, 2000

8
Modeling Circles
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An Interpolatory Scheme for Circles

• Use a different set of interpolating functions to
  compute weights for new vertices
• Solve for weights like before
• Capable of reproducing global functions
• represent circles

10
Form of the Weights

• Weights depend on level of subdivision
• Limit is of non-stationary scheme is
• Dyn and Levin, 1995

11
Geometric Interpretation of Weights

• is a tension associated with subdivision
  scheme
• Tensions determine how much the curve pulls away
  from edges of original polygon
• To produce a circle choose to be

12
Factoring the Subdivision Step

• Factor into linear subdivision followed by
  differencing

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The Differencing Mask

• Linear subdivision isolates the addition of new
  vertices
• Differencing repositions vertices
• Rule is uniform

14
Extension to Surfaces

• Linear subdivision Bilinear subdivision
• Differencing Two-dimensional differencing
• Use tensor product

15
Surface Example

• Linear subdivision Differencing
• Subdivision method for curve networks

16
Example Circular Torus

• Tensions set to zero to produce a circle

17
Cylinder Example

• Open boundary converges to a circle as well

18
Extensions

• Open meshes
• Extraordinary vertices
• Non-manifold geometry
• Tagged meshes for creases

19
Demo

• Construct profile curve to define surfaces of
  revolution

20
Conclusions

• Developed curve scheme to produce circles
• Tensions control shape of the curve
• Factored subdivision into linear subdivision plus
  differencing
• Extended to surfaces