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 Catmull-Clark subdivision surface (CCSS) initial mesh step 1limit surface

CCSS patch: regular vs. extraordinary  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 Control meshLimit surface Blue: regular Red: extraordinary

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

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,

Limit mesh approximation  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  Limit mesh : push the control points to their limit positions.  It inscribes the limit surface

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

Transformation into bicubic Bézier forms  Both and can be transformed into bicubic Bézier form :

Regular patches: distance bound  Core idea: Measure through measuring

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,

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]

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

Extraordinary patches: distance bound  Limit face can be partitioned into bilinear subfaces defined over :  Similar to the regular case, for  By solving 16 constrained minimization problems, we have

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,

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  Only needed to consider 2 subpatches and

Extraordinary patches: bound constant , , strictly decreases as n increases

Comparison of bound constants  First two lines are for control mesh approx.  Last line are for limit mesh approximation , n Cheng et al Huang et al Limit mesh

Application to adaptive subdivison Error tolerance Frog modelCar model Control meshLimit meshControl meshLimit mesh 0.125,64214,52714,7178, ,86426,48224,90514, ,159113,71397,70458,805 The number of faces decreases by about 30%

Application to CCSS intersection

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

Thank you!