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NURBS with Extraordinary Points: Highdegree, Nonuniform, Rational Subdivision Schemes
"... The same control mesh subdivided in three different configurations. In regular regions, the degree 3 surface (b) is C2, and the degree 7 surfaces (c) and (d) are C6. No previous subdivision scheme can generate the surface (d). The knot intervals modified to give (d) are shown in (a); the red interva ..."
Abstract

Cited by 14 (8 self)
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The same control mesh subdivided in three different configurations. In regular regions, the degree 3 surface (b) is C2, and the degree 7 surfaces (c) and (d) are C6. No previous subdivision scheme can generate the surface (d). The knot intervals modified to give (d) are shown in (a); the red interval is ten times greater than the unmarked intervals, and the green interval is four times greater. Comparing (d) with (c), note that in this case the nonuniform intervals change the whole surface, because the influence of a knot interval grows with degree. We present a subdivision framework that adds extraordinary vertices to NURBS of arbitrarily high degree. The surfaces can represent any odd degree NURBS patch exactly. Our rules handle nonuniform knot vectors, and are not restricted to midpoint knot insertion. In the absence of multiple knots at extraordinary points, the limit surfaces have bounded curvature.
Subdivision Surface Based Onepiece Representation
, 2006
"... Subdivision surfaces are capable of modeling and representing complex shape of arbitrary topology. However, methods on how to build the control mesh of a complex surface are not studied much. Currently, most meshes of complicated objects come from triangulation and simplification of raster scanned ..."
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Subdivision surfaces are capable of modeling and representing complex shape of arbitrary topology. However, methods on how to build the control mesh of a complex surface are not studied much. Currently, most meshes of complicated objects come from triangulation and simplification of raster scanned data points, like the Stanford 3D Scanning Repository. This approach is costly and leads to very dense meshes. Subdivision surface based onepiece representation means to represent the final object in a design process with only one subdivision surface (i.e. a sparse control mesh), no matter how complicated the object’s topology or shape. No decomposition of the object into simpler components is necessary. Hence the number of parts in the final representation is always the minimum: one. We have developed necessary mathematical theories and geometric algorithms to support subdivision surface based onepiece representation. First an explicit parametrization method is presented for exact evaluation of CatmullClark subdivision surfaces. Based on our parametrization techniques, two approaches have been proposed for constructing a control mesh of a given object with arbitrary topology. The first approach
Tessellation, Fairing, Shape Design, and Trimming Techniques for Subdivision Surface based Modeling
, 2008
"... Subdivision surfaces are capable of modeling and representing complex shape of arbitrary topology. However, methods on how to build the control mesh of a complex surface have not been studied much. Currently, most meshes of complicated objects come from triangulation and simplification of raster sca ..."
Abstract
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Subdivision surfaces are capable of modeling and representing complex shape of arbitrary topology. However, methods on how to build the control mesh of a complex surface have not been studied much. Currently, most meshes of complicated objects come from triangulation and simplification of raster scanned data points, like the Stanford 3D Scanning Repository. This approach is costly and leads to very dense meshes. In this project, we develop necessary mathematical theories and geometric algorithms to support subdivision surface based modeling. First, an explicit parametrization method is presented for exact evaluation of CatmullClark subdivision surfaces. Based on our parametrization techniques, two approaches have been developed for constructing a control mesh of a given object with arbitrary topology. The first approach is interpolation. By sampling some representative points from a given object model, a control mesh can be constructed and its subdivision surface interpolates all the sampled representative points and meanwhile is very close to the given data model. Interpolation is a simple way to build models, but the fairness of the interpolating surface is a big concern in previous methods. By using similarity based interpolation, we can obtain better modeling result with less undesired artifacts and
Tessellation, Fairing, Shape Design, and Trimming Techniques for Subdivision Surface based Modeling
, 2008
"... Subdivision surfaces are capable of modeling and representing complex shape of arbitrary topology. However, methods on how to build the control mesh of a complex surface have not been studied much. Currently, most meshes of complicated objects come from triangulation and simplification of raster sca ..."
Abstract
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Subdivision surfaces are capable of modeling and representing complex shape of arbitrary topology. However, methods on how to build the control mesh of a complex surface have not been studied much. Currently, most meshes of complicated objects come from triangulation and simplification of raster scanned data points, like the Stanford 3D Scanning Repository. This approach is costly and leads to very dense meshes. In this project, we develop necessary mathematical theories and geometric algorithms to support subdivision surface based modeling. First, an explicit parametrization method is presented for exact evaluation of CatmullClark subdivision surfaces. Based on our parametrization techniques, four modeling techniques have been developed for constructing the control mesh of a given object with arbitrary topology. The first technique is adaptive tessellation. By performing recursive adaptive subdivision of an initial mesh, we can get the control mesh of a given object with the same geometric appearance and properties but with much less nodes in the control mesh of the object. This work needs the capability of estimating the subdivision depth of the given object within a given error tolerance before hand. The subdivision depth computation technique is presented in this report.