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Subdivision for Modeling and Animation
 SIGGRAPH ’99 Courses, no. 37. ACM SIGGRAPH
, 1999
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Analysis of Algorithms Generalizing BSpline Subdivision
 SIAM J. Numer. Anal
, 1997
"... A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algorithms is presented. The main challenge here is the verification of injectivity of the characteristic map. The tools are sufficiently versatile and easy to wield to allow, as an application, a full analys ..."
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A new set of tools for verifying smoothness of surfaces generated by stationary subdivision algorithms is presented. The main challenge here is the verification of injectivity of the characteristic map. The tools are sufficiently versatile and easy to wield to allow, as an application, a full analysis of algorithms generalizing biquadratic and bicubic Bspline subdivision. In the case of generalized biquadratic subdivision the analysis yields a hitherto unknown sharp bound strictly less than one on the second largest eigenvalue of any smoothly converging subdivision. Keywords: subdivision, arbitrary topology, characteristic map, DooSabin Algorithm, CatmullClark algorithm, Bspline AMS subject classification: 65D17, 65D07, 68U07 Abbreviated title: Generalized BSpline Subdivision 1 Introduction The idea of generating smooth freeform surfaces of arbitrary topology by iterated mesh refinement dates back to 1978, when two papers [CC78], [DS78] appeared back to back in the same issue ...
Improved Triangular Subdivision Schemes
, 1998
"... In this article we improve the butterfly and Loop's algorithm. As a result we obtain subdivision algorithms for triangular nets which can be used to generate G 1  and G 2  surfaces, respectively. Keywords: Subdivision, interpolatory subdivision, Loop's algorithm, butterfly algorithm ..."
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In this article we improve the butterfly and Loop's algorithm. As a result we obtain subdivision algorithms for triangular nets which can be used to generate G 1  and G 2  surfaces, respectively. Keywords: Subdivision, interpolatory subdivision, Loop's algorithm, butterfly algorithm. 1. Introduction Subdivision algorithms are popular in CAGD since they provide simple, efficient tools to generate arbitrary free form surfaces. For example, the algorithms by Catmull and Clark [3] and Loop [7] are generalizations of wellknown spline subdivision schemes. Therefore the surfaces produced by these algorithms are piecewise polynomial and at ordinary points curvature continuous. At extraordinary points however, the curvature is zero or infinite. In general, singularities at extraordinary points is an inherent phenomenon of subdivision, see [13, 12, 9]. The smoothness of a subdivision surface at its extraordinary points depends on the spectral properties of the associated subdivision mat...
Smoothness of stationary subdivision on irregular meshes
 Constructive Approximation
, 1998
"... We derive necessary and sufficient conditions for tangent plane and C kcontinuity of stationary subdivision schemes near extraordinary vertices. Our criteria generalize most previously known conditions. We introduce a new approach to analysis of subdivision surfaces based on the idea of the univers ..."
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We derive necessary and sufficient conditions for tangent plane and C kcontinuity of stationary subdivision schemes near extraordinary vertices. Our criteria generalize most previously known conditions. We introduce a new approach to analysis of subdivision surfaces based on the idea of the universal surface. Any subdivision surface can be locally represented as a projection of the universal surface, which is uniquely defined by the subdivision scheme. This approach provides us with a more intuitive geometric understanding of subdivision near extraordinary vertices. AMS MOS classification: 65D10, 65D17, 68U05
Triangle mesh subdivision with bounded curvature and the convex hull property
"... The masks for Loop’s triangle subdivision surface algorithm are modified resulting in surfaces with bounded curvature and the convex hull property. New edge masks are generated by a cubic polynomial mask equation whose Chebyshev coefficients are closely related to the eigenvalues of the correspond ..."
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The masks for Loop’s triangle subdivision surface algorithm are modified resulting in surfaces with bounded curvature and the convex hull property. New edge masks are generated by a cubic polynomial mask equation whose Chebyshev coefficients are closely related to the eigenvalues of the corresponding subdivision matrix. The mask equation is found to satisfy a set of smoothness constraints on these eigenvalues. We observe that controlling the root structure of the mask equation is important for deriving subdivision masks with nonnegative weights.
Subdivision Surfaces  Can they be Useful for Geometric Modeling Applications?
, 2001
"... This report summarizes the findings and recommendations of the authors concerning the usefulness of subdivision surfaces for geometric modeling, and in particular for engineering applications. The work described is a result of a threemonth collaboration of the authors during the visit of the sec ..."
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This report summarizes the findings and recommendations of the authors concerning the usefulness of subdivision surfaces for geometric modeling, and in particular for engineering applications. The work described is a result of a threemonth collaboration of the authors during the visit of the second author to Boeing in the Summer of 2001.
Convex triangular subdivision surfaces with bounded curvature
, 2000
"... The edge masks for Loop’s triangular subdivision surface algorithm are modified resulting in surfaces with bounded curvature and the convex hull property. The new edge masks are derived from a polynomial mask equation whose Chebyshev expansion coefficients are closely related to the eigenvalues of t ..."
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The edge masks for Loop’s triangular subdivision surface algorithm are modified resulting in surfaces with bounded curvature and the convex hull property. The new edge masks are derived from a polynomial mask equation whose Chebyshev expansion coefficients are closely related to the eigenvalues of the corresponding subdivision matrix. The mask equation is found to satisfy a set of smoothness constraints on these eigenvalues. 1
Fast evaluation of the improved loop’s subdivision surfaces
 Proc. of the Geometric Modeling and Processing 2004. Beijing: IEEE Computer Society
, 2004
"... For obtaining better curvature distributions of subdivision surfaces, various improvements for Loop’s subdivision scheme on triangular surface meshes have been made. A careful analysis shows that the fast evaluation technique of the subdivision surface proposed by Stam is no longer usable to these ..."
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For obtaining better curvature distributions of subdivision surfaces, various improvements for Loop’s subdivision scheme on triangular surface meshes have been made. A careful analysis shows that the fast evaluation technique of the subdivision surface proposed by Stam is no longer usable to these improved schemes. This paper describes a fast evaluating algorithm for improved Loop’s subdivision surfaces. The algorithm is applicable to a vast class of subdivision schemes for triangular surface meshes. Using the proposed algorithm, one can evaluate the subdivision surface at any domain point for any set of input subdivision masks and a control mesh.
MASS: MULTIRESOLUTIONAL ADAPTIVE SOLID SUBDIVISION
, 2003
"... In this survey paper, we discuss subdivision geometry, subdivision schemes, its analysis and applications, especially from the view of solid modeling. Subdivision technique has been widely accepted in computer graphics and geometric design applications. However, it has been largely ignored in solid ..."
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In this survey paper, we discuss subdivision geometry, subdivision schemes, its analysis and applications, especially from the view of solid modeling. Subdivision technique has been widely accepted in computer graphics and geometric design applications. However, it has been largely ignored in solid modeling. The first few sections are devoted to the history of subdivision modeling and the review of existing subdivision schemes in detail. We also briefly review other solid modeling techniques. Next, we discuss the current mathematical technique to analyze subdivision schemes on both regular and extraordinary topologies. We provide examples of analysis on the schemes in prior sections. We discuss problems involving solid scheme analysis and suggest possible solutions. In addition, we review prior work using subdivision technique in various applications. The latter part of the paper devote to our novel subdivision solid schemes, ongoing research topics, and new ideas. We demonstrate the potential of subdivision solids by a variety of examples. Attractive features of subdivision solids are compared and addressed. We conclude the paper with the summary and the expectation for future of subdivision technique in solid modeling.