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A Factored, Interpolatory Subdivision for Surfaces of Revolution

A Factored, Interpolatory Subdivision for Surfaces of Revolution. Scott Schaefer Joe Warren. Rice University. Importance of Subdivision. Allows coarse, low-polygon models to approximate smooth shapes. Subdivision.

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A Factored, Interpolatory Subdivision for Surfaces of Revolution

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  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. f (t) 4 3.5 3 2.5 2 9 16 ___ 1.5 1 0.5 t 0.5 1 1.5 2 2.5 3 -1 16 -1 16 ___ ___ 9 16 ___ 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

  9. 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. f (t) 4 3.5 3 2.5 2 1.5 -w 16 __ n 1 0.5 t 0.5 1 1.5 2 2.5 3 8+w 16 8+w 16 -w 16 ___ ___ __ n n n 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

  13. 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

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