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√3subdivision
 IN PROCEEDINGS OF ACM SIGGRAPH
, 2000
"... A new stationary subdivision scheme is presented which performs slower topological refinement than the usual dyadic split operation. The number of triangles increases in every step by a factor of 3 instead of 4. Applying the subdivision operator twice causes a uniform refinement with trisection of ..."
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Cited by 138 (4 self)
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A new stationary subdivision scheme is presented which performs slower topological refinement than the usual dyadic split operation. The number of triangles increases in every step by a factor of 3 instead of 4. Applying the subdivision operator twice causes a uniform refinement with trisection of every original edge (hence the name 3subdivision) while two dyadic splits would quadsect every original edge. Besides the finer gradation of the hierarchy levels, the new scheme has several important properties: The stencils for the subdivision rules have minimum size and maximum symmetry. The smoothness of the limit surface is C2 everywhere except for the extraordinary points where it is C1. The convergence analysis of the scheme is presented based on a new general technique which also applies to the analysis of other subdivision schemes. The new splitting operation enables locally adaptive refinement under builtin preservation of the mesh consistency without temporary crackfixing between neighboring faces from different refinement levels. The size of the surrounding mesh area which is affected by selective refinement is smaller than for the dyadic split operation. We further present a simple extension of the new subdivision scheme which makes it applicable to meshes with boundary and allows us to generate sharp feature lines.
Subdivision Surfaces: A New Paradigm For ThinShell FiniteElement Analysis
 INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING
, 2000
"... We develop a new paradigm for thinshell finiteelement analysis based on the use of subdivision surfaces for: i) describing the geometry of the shell in its undeformed configuration, and ii) generating smooth interpolated displacement fields possessing bounded energy within the strict framework ..."
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Cited by 133 (30 self)
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We develop a new paradigm for thinshell finiteelement analysis based on the use of subdivision surfaces for: i) describing the geometry of the shell in its undeformed configuration, and ii) generating smooth interpolated displacement fields possessing bounded energy within the strict framework of the KirchhoffLove theory of thin shells. The particular subdivision strategy adopted here is Loop's scheme, with extensions such as required to account for creases and displacement boundary conditions. The displacement fields obtained by subdivision are H 2 and, consequently, have a finite KirchhoffLove energy. The resulting finite elements contain three nodes and element integrals are computed by a onepoint quadrature. The displacement field of the shell is interpolated from nodal displacements only. In particular, no nodal rotations are used in the interpolation. The interpolation scheme induced by subdivision is nonlocal, i. e., the displacement field over one element depend on the nodal displacements of the element nodes and all nodes of immediately neighboring elements. However, the use of subdivision surfaces ensures that all the local displacement fields thus constructed combine conformingly to define one single limit surface.
A sketchbased interface for detailpreserving mesh editing
 ACM Trans. Graph
, 2005
"... In this paper we present a method for the intuitive editing of surface meshes by means of viewdependent sketching. In most existing shape deformation work, editing is carried out by selecting and moving a handle, usually a set of vertices. Our system lets the user easily determine the handle, eit ..."
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Cited by 91 (6 self)
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In this paper we present a method for the intuitive editing of surface meshes by means of viewdependent sketching. In most existing shape deformation work, editing is carried out by selecting and moving a handle, usually a set of vertices. Our system lets the user easily determine the handle, either by silhouette selection and cropping, or by sketching directly onto the surface. Subsequently, an edit is carried out by sketching a new, viewdependent handle position or by indirectly influencing differential properties along the sketch. Combined, these editing and handle metaphors greatly simplify otherwise complex shape modeling tasks.
4–8 Subdivision
, 2000
"... In this paper we introduce 4–8 subdivision, a new scheme that generalizes the fourdirectional box spline of class � � to surfaces of arbitrary topological type. The crucial advantage of the proposed scheme is that it uses bisection refinement as an elementary refinement operation, rather than more c ..."
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Cited by 66 (6 self)
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In this paper we introduce 4–8 subdivision, a new scheme that generalizes the fourdirectional box spline of class � � to surfaces of arbitrary topological type. The crucial advantage of the proposed scheme is that it uses bisection refinement as an elementary refinement operation, rather than more commonly used face or vertex splits. In the uniform case, bisection refinement results in doubling, rather than quadrupling of the number of faces in a mesh. Adaptive bisection refinement, automatically generates conforming variableresolution meshes in contrast to face and vertex split methods which require an postprocessing step to make an adaptively refined mesh conforming. The fact that the size of faces decreases more gradually with refinement allows one to have greater control over the resolution of a refined mesh. It also makes it possible to achieve higher smoothness while using small stencils (the size of the stencils used by our scheme is similar to Loop subdivision). We show that the subdivision surfaces produced by the 4–8 scheme are � � continuous almost everywhere, except at extraordinary vertices where they are is �continuous.
A Unified Framework for Primal/Dual Quadrilateral Subdivision Schemes
 CAGD
, 2001
"... Quadrilateral subdivision schemes come in primal and dual varieties, splitting faces or respectively vertices. The scheme of CatmullClark is an example of the former, while the DooSabin scheme exemplifies the latter. In this paper we consider the construction of an increasing sequence of alternati ..."
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Cited by 42 (4 self)
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Quadrilateral subdivision schemes come in primal and dual varieties, splitting faces or respectively vertices. The scheme of CatmullClark is an example of the former, while the DooSabin scheme exemplifies the latter. In this paper we consider the construction of an increasing sequence of alternating primal/dual quadrilateral subdivision schemes based on a simple averaging approach. Beginning with a vertex split step we successively construct variants of DooSabin and CatmullClark schemes followed by novel schemes generalizing Bsplines of bidegree up to nine. We prove the schemes to be C¹ at irregular surface points, and analyze the behavior of the schemes as the number of averaging steps increases. We discuss a number of implementation issues common to all quadrilateral schemes. In particular we show how both primal and dual quadrilateral schemes can be implemented in the same code, opening up new possibilities for more flexible geometric modeling applications and pversions of the Subdivision Element Method. Additionally we describe a simple algorithm for adaptive subdivision of dual schemes.
Approximate Boolean Operations on Freeform Solids
, 2001
"... In this paper we describe a method for computing approximate results of boolean operations (union, intersection, difference) applied to freeform solids bounded by multiresolution subdivision surfaces. ..."
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Cited by 41 (5 self)
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In this paper we describe a method for computing approximate results of boolean operations (union, intersection, difference) applied to freeform solids bounded by multiresolution subdivision surfaces.
Fully C1Conforming Subdivision Elements for Finite Deformation ThinShell Analysis
, 2001
"... We have extended the subdivision shell elements of Cirak et al. [18] to the finitedeformation range. The assumed finitedeformation kinematics allows for finite membrane and thickness stretching, as well as for large deflections and bending strains. The interpolation of the undeformed and deformed ..."
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Cited by 40 (14 self)
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We have extended the subdivision shell elements of Cirak et al. [18] to the finitedeformation range. The assumed finitedeformation kinematics allows for finite membrane and thickness stretching, as well as for large deflections and bending strains. The interpolation of the undeformed and deformed surfaces of the shell is accomplished through the use of subdivision surfaces. The resulting ‘subdivision elements’ are strictly C1conforming, contain three nodes and one single quadrature point per element, and carry displacements at the nodes only. The versatility and good performance of the subdivision elements is demonstrated with the aid of a number of test cases, including the stretching of a tension strip; the inflation of a spherical shell under internal pressure; the bending and inflation of a circular plate under the action of uniform pressure; and the inflation of square and circular airbags. In particular, the airbag solutions, while exhibiting intricate folding patterns, appear to converge in certain salient features of the solution, which attests to the robustness of the method.
Linear anisotropic mesh filtering
 IBM Research Report RC22213(W0110051), IBM T.J. Watson Research
, 2001
"... This report has been submitted for publication outside of IBM and will probably be copyrighted is accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM ..."
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Cited by 40 (1 self)
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This report has been submitted for publication outside of IBM and will probably be copyrighted is accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and specific requests. After outside publication, requests should be filled only by reprints or legally obtained copies of the article (e.g., payment of royalties). Some reports are available at