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Unformatted text preview: Adjustable Speed Surface Subdivision K. Karciauskas and J. Peters Department of Mathematics and Informatics, Vilnius University Dept C.I.S.E., University of Florida Abstract We introduce a non-uniform subdivision algorithm that partitions the neighborhood of an extraordinary point in the ratio : 1 , where (0 , 1) . We call the speed of the non-uniform subdivision and verify C 1 continuity of the limit surface. For = 1 / 2 , the Catmull-Clark limit surface is recovered. Other speeds are useful to vary the relative width of the polynomial spline rings generated from extraordinary nodes. Key words: non-uniform, subdivision, adjustable speed, = 1 2 = 3 4 = 7 8 = 1 2 = 3 4 = 7 8 Fig. 1. ( left ) Input with gap and ( right three columns) fillings with ( top ) one, respectively ( bottom ) two adjustable speed subdivision steps. Email address: email@example.com,firstname.lastname@example.org (K. Karciauskas and J. Peters). URL: http://www.cise.ufl.edu/ jorg (K. Karciauskas and J. Peters). Preprint submitted to Dec 2007 1 Introduction To be able to control and adjust, possibly repeatedly, the relative width of the poly- nomial surface rings generated by a Catmull-Clark-like subdivision algorithm (see Figure 1), we derive rules for a new class of non-uniform surface subdivision al- gorithms. The resulting family of algorithms has properties akin to Catmull-Clark subdivision and can in particular generate the limit surface of Catmull-Clark sub- division. Figure 8 shows an application where fast filling of a n-sided surface hole is advantageous. In general, the fill speed must depend on the application since the surface quality deteriorates for more rapid fills. We have also used the characteristic rings of adjustable speed subdivision as concentric tessellation map of guided sub- division surfaces [KP07b], where the speed controls the parameterization rather than the shape (see Figure 9). The theory is advanced by the paper in that it ex- plores a non-traditional approach that allows analyzing the resulting non-uniform subdivision surface. Figure 2 illustrates adjustable speed subdivision in one vari- = 1 2 = 2 3 Fig. 2. Univariate localized subdivision near a (central) point, ( left ) in the standard setting and ( right ) adjustable speed subdivision (local truncated geometric progression per half space) with speed = 2 / 3 . ( top two) control point refinement, ( bottom ) knot spacing. able. Instead of subdividing uniformly, we locally subdivide in a ratio : where (0 , 1) and := 1 . We call the speed of the non-uniform subdivision. Since splines are invariant when scaling all knots uniformly or when translating the knot sequence, only the knot spacing is in the following of interest....
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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.
- Spring '08