1. Scott Schaefer Joe Warren A Factored, Interpolatory Subdivision for Surfaces of Revolution Rice University

2. Allows coarse, low-polygon models to approximate smooth shapes Allows coarse, low-polygon models to approximate smooth shapes Importance of Subdivision

3. Subdivision A process that takes a polygon as input and produces a new polygon as output A process that takes a polygon as input and produces a new polygon as output Defines a sequence which should converge in the limit 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 Subdivision scheme is interpolatory if the vertices of are a subset of the verticesof Example: linear subdivision Example: linear subdivision

5. Interpolatory Scheme Place new point on curve defined by a cubic interpolant through 4 consecutive points 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 If parameterization is uniform, weights do not depend on scale

6. Curve Subdivision Example Produces a curve that is Produces a curve that is Cannot reproduce circles Cannot reproduce circles

7. Extension to Surfaces Extended to quadrilateral surfaces of arbitrary topology [Kobbelt, 1995] Extended to quadrilateral surfaces of arbitrary topology [Kobbelt, 1995] Surface subdivision scheme is Surface subdivision scheme is [Zorin, 2000]

8. Modeling Circles

9. An Interpolatory Scheme for Circles Use a different set of interpolating functions to compute weights for new vertices Use a different set of interpolating functions to compute weights for new vertices Solve for weights like before Solve for weights like before Capable of reproducing global functions Capable of reproducing global functions represent circles represent circles

10. 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]

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

12. Factoring the Subdivision Step Factor into linear subdivision followed by differencing Factor into linear subdivision followed by differencing

13. The Differencing Mask Linear subdivision isolates the addition of new vertices Linear subdivision isolates the addition of new vertices Differencing repositions vertices Differencing repositions vertices Rule is uniform Rule is uniform

14. Extension to Surfaces Linear subdivision Bilinear subdivision Linear subdivision Bilinear subdivision Differencing Two-dimensional differencing Differencing Two-dimensional differencing Use tensor product Use tensor product

15. Surface Example Linear subdivision + Differencing Linear subdivision + Differencing Subdivision method for curve networks Subdivision method for curve networks

16. Example: Circular Torus Tensions set to zero to produce a circle Tensions set to zero to produce a circle

17. Cylinder Example Open boundary converges to a circle as well Open boundary converges to a circle as well

18. Extensions Open meshes Open meshes Extraordinary vertices Extraordinary vertices Non-manifold geometry Non-manifold geometry Tagged meshes for creases Tagged meshes for creases

19. Demo Construct profile curve to define surfaces of revolution Construct profile curve to define surfaces of revolution

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