file_d41d8cd98f00b204e9800998ecf8427e - Adjustable Speed...

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Adjustable Speed Subdivision Surfaces K. Karˇciauskas and J. Peters Abstract —We introduce a non-uniform subdivision algorithm that partitions the neighborhood of an extraordinary point in the ratio σ : 1 - σ , where σ (0 , 1) . We call σ the speed of the non- uniform subdivision and verify C 1 continuity of the limit surface. For σ = 1 / 2 , the Catmull-Clark algorithm is recovered. Other speeds are useful to vary the contraction near extraordinary points. I. INTRODUCTION With few exceptions, polynomial subdivision algorithms generalize uniform (box-)spline subdivision [BHR93]. In the univariate case, standard B-spline knot insertion yields a stable, local non-uniform refinement of the control polygon, one knot at a time. Cashman et al. [CDS07], [CDS08] recently proposed factoring non-uniform B-spline subdivision into small stencils suitable for simultaneous knot insertion. Earlier, Goldman and Warren [GW93] extended uniform subdivision of curves to knot intervals in globally geometric progression. When the univariate refinement is tensored and applied to tensor-product splines, the knot intervals must be carried over to parallel iso-curves. In practice, the least superset of all iso-knot intervals is enforced by inserting knots. By con- trast, Non-uniform Recursive Subdivision Surfaces, proposed by Sederberg et al. [SZSS98], allow freely assigning knot intervals to every edge of the control net. Subdivision in the regular part surrounding an extraordinary node, i.e. a control point associating more or fewer than four direct neighbors, is constructed by halving each knot interval. blend with hole Fig. 1. Shape completion. ( top ) Regular speed σ = 1 / 2 . ( bottom ) High speed σ = 3 / 4 . ( left ) Blend with hole. ( middle ) Two steps of subdivision. ( right ) Highlight lines when filling the remaining hole according to [SS05]. Note the strong fault in the highlight line in the center area of the zoomed in detail ( top , right ). We came across a different generalization, not covered by [GW93], [SZSS98], [CDS07], when looking for reparameter- izations for high-quality surface constructions. Our approach is philosophically different. We take the point of view that subdivision surfaces are splines with singularities and that subdivision is a process for generating a sequence of nested surface rings. Once the Bernstein-B´ezier (BB-) form of a surface ring is well-defined, e.g. as in Appendix V-C, we do not refine the control net further, but work with the polynomial representation. In this scenario, the non-uniformity is motivated by the the desire to adjust, possibly repeatedly, the relative width of polynomial surface rings. Figure 1 gives an example of such an adjustment in practice. The resulting family of algorithms has properties akin to and specializes to Catmull-Clark subdivision. In Section II, we derive the rules and in Section III, we verify the properties. Besides adjustable control net refinement for computer graphics and adjustable-
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This note was uploaded on 01/20/2012 for the course CIS 4914 taught by Professor Desimone during the Spring '08 term at University of Florida.

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file_d41d8cd98f00b204e9800998ecf8427e - Adjustable Speed...

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