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Proceedings of CAD'14, 2014, 222-224
T-Spline Polygonal Complexes

Abdulwahed M. Abbas, The University of Balamand

Abstract. The earliest notion of polygonal (strip) complexes was proposed by Nasri as a means of interpolating uniform quadratic B-spline curves by Doo-Sabin subdivision surfaces. Later on, Nasri redeployed this notion to also support the interpolation of uniform cubic B-spline curves by Catmull-Clark subdivision surfaces. The initial motivation behind a polygonal complex is that, under the corresponding subdivision scheme, it admits a B-spline limit curve of the same degree. Thus, when a complex is embedded within a polygonal mesh, its limit curve is automatically interpolated by the surface limit of subdivision of the polygonal mesh, without the need for any additional overheads. More interestingly nothing prevents the extension of this notion to any other subdivision scheme. This notion was employed later, under well-specified constraints, for the interpolation of any arbitrarily intersecting network of curves by a subdivision surface. This works for Doo-Sabin and Catmull-Clark as well as for Loop subdivision schemes. The main goal of the paper is to be able to interpolate B-spline curves by T-spline surfaces directly through the application of a single mathematical formula in much the same way that has been successfully done in the context of Doo-Sabin, and Catmull-Clark Subdivision Surfaces and later on in the context of Loop subdivision Surfaces. To this end, the paper proceeds from the basic definition of B-spline curves and surfaces and managed to derive a formulation of Polygonal Complexes for B-spline surfaces.

Keywords. B-splines, polygonal complexes, subdivision surfaces, NURBS, T-spline surfaces

DOI: 10.14733/cadconfP.2014.222-224