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Distance Between a Catmull-Clark Subdivision Surface and Its Limit Mesh

Distance Between a Catmull-Clark Subdivision Surface and Its Limit Mesh. Zhangjin Huang, Guoping Wang Peking University, China. Generalization of uniform bicubic B-spline surface continuous except at extraordinary points The limit of a sequence of recursively refined control meshes.

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Distance Between a Catmull-Clark Subdivision Surface and Its Limit Mesh

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  1. Distance Between a Catmull-Clark Subdivision Surface and Its Limit Mesh Zhangjin Huang, Guoping Wang Peking University, China

  2. Generalization of uniform bicubic B-spline surface continuous except at extraordinary points The limit of a sequence of recursively refined control meshes Catmull-Clark subdivision surface (CCSS) initial mesh step 1 limit surface

  3. CCSS patch: regular vs. extraordinary Blue: regular Red: extraordinary Control mesh Limit surface • Assume each mesh face in the control mesh • a quadrilateral • at most one extraordinary point (valence n is not 4) • An interior mesh face in the control mesh → a surface patch in the limit surface • Regular patch: bicubic B-spline patches, 16 control points • Extraordinary patch: not B-spline patches, 2n+8 control points

  4. Control mesh approximation and error Control mesh is a piecewise linear approximation to a CCSS Approximation error: the maximal distance between a CCSS and the control mesh • Distance between a CCSS patch and its mesh face (or control mesh) is defined as • is unit square • is Stam’s parametrization of over • is bilinear parametrization of over

  5. Distance bound for control mesh approximation • The distance between a CCSS patch and its control mesh is bounded as [Cheng et al. 2006] • is a constant that only depends on valence n • is the the second order norm of 2n+8 control points of • For regular patches,

  6. Limit mesh approximation • Limit mesh: push the control points to their limit positions. • It inscribes the limit surface • An interior mesh face → a limit face → a surface patch • We derive a bound on the distance between a patch and its limit face (or limit mesh) as • means that the limit mesh approximates a CCSS better than the control mesh

  7. Regular patches: how to estimate • Regular patch is a bicubic B-spline patch: • Limit face,then • It is not easy to estimate directly!

  8. Transformation into bicubic Bézier forms • Both and can be transformed into bicubic Bézier form:

  9. Regular patches: distance bound • Core idea: Measure through measuring

  10. Regular patches: distance bound (cont.) • Bound with the second order norm , it follows that • Distance bound function of with respect to is • Diagonal • By symmetry,

  11. Regular patch: distance bound (cont.) • Theorem 1 The distance between a regular CCSS patch and the corresponding limit face is bounded by • The distance between a regular patch and its corresponding mesh face is bounded as [Cheng et al 2006]

  12. Extraordinary patches: parametrization • An extraordinary patch can be partitioned into an infinite sequence of uniform bicubic B-spline patches • Partition the unit square into tiles • Stam’s parametrization: • Transformation maps the tile onto the unit square

  13. Extraordinary patches: distance bound • Limit facecan bepartitioned into bilinear subfaces defined over : • Similar to the regular case, for • By solving 16 constrained minimization problems, we have

  14. Extraordinary patch: distance bound function • Thus • is the distance bound function of with respect to : • The distance bound function of with respect to is defined as: • Diagonal • By symmetry,

  15. Extraordinary patches: distance bound • Theorem 2 The distance between an extra-ordinary CCSS patch and the corresponding limit face is bounded by • has the following properties: • , attains its maximum in • , attains its maximum in • Only needed to consider 2 subpatches and

  16. Extraordinary patches: bound constant • , • , strictly decreases as n increases

  17. Comparison of bound constants • First two lines are for control mesh approx. • Last line are for limit mesh approximation • , • ,

  18. Application to adaptive subdivison The number of faces decreases by about 30%

  19. Application to CCSS intersection

  20. Conclusion • Propose an approach to derive a bound on the distance between a CCSS and its limit mesh • Our approach can be applied to other spline based subdivision surfaces • Show that a limit mesh may approximate a CCSS better than the corresponding control mesh

  21. Thank you!

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