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Splines and geometric modeling [chapter]

Chandrajit Bajaj
2004 Handbook of Discrete and Computational Geometry, Second Edition  
Storry and Ball SB89 demonstrate that a B spline subdivision patch can be tted into a general n,sided area of a bicubic surface with at least tangent plane continuity on the boundary.  ...  The subdivision scheme developed by Loop Loo87 splits each triangle of a triangular mesh into four triangles.  ... 
doi:10.1201/9781420035315.ch53 fatcat:e7evq2b77ba5lj3vxlcxwxxzda

Blending, smoothing and interpolation of irregular meshes using N-sided Varady patches

Xuefu Wang, Fuhua (Frank) Cheng, Brian A. Barsky
1999 Proceedings of the fifth ACM symposium on Solid modeling and applications - SMA '99  
the hole filling process and having an additional degree of freedom in adjusting the shape of an n-sided patch); (3) overall smoothness and more even distribution of curvature because of the global energy  ...  N-sided krady surface representation is not new, but its use in solving modeling problems has been limited.  ...  For the vertex blending problem, one usually needs to decompose the vertex blending region (usually an n-sided region) into triangular or rectangular sub-regions.  ... 
doi:10.1145/304012.304034 dblp:conf/sma/WangCB99 fatcat:dsk7bjszzzdk7bzwj5xwx6tddi

Pattern-Based Quadrangulation forN-Sided Patches

Kenshi Takayama, Daniele Panozzo, Olga Sorkine-Hornung
2014 Computer graphics forum (Print)  
Our algorithm is guaranteed to successfully quadrangulate N-sided patches with arbitrary numbers of edge subdivisions specified at the patch boundary.  ...  Abstract We propose an algorithm to quadrangulate an N-sided patch (2 ≤ N ≤ 6) with prescribed numbers of edge subdivisions at its boundary.  ...  The Bear model is courtesy of Takeo Igarashi. The Humanoid model is courtesy of the AIM@SHAPE Shape Repository. This work was supported in part by the ERC grant iModel (StG-2012-306877).  ... 
doi:10.1111/cgf.12443 fatcat:cd7negr2ebfyznwz3oog4og6rm

CrossFill: Foam Structures with Graded Density for Continuous Material Extrusion

Tim Kuipers, Jun Wu, Charlie C.L. Wang
2019 Computer-Aided Design  
Dithering of the subdivision levels is performed to accurately reproduce a prescribed density distribution.  ...  Our method for generating CrossFill is based on a space-filling surface that employs spatially varying subdivision levels.  ...  The subdivision tiling is generated by subdividing an H-prism into two Q-prisms and subdividing a Q-prism into four H-prisms (see the bottom row of Fig. 5 ).  ... 
doi:10.1016/j.cad.2019.05.003 fatcat:lc6q3spjcbdmtcbdutw5bxarui

Pairs of bi-cubic surface constructions supporting polar connectivity

Ashish Myles, Kęstutis Karčiauskas, Jörg Peters
2008 Computer Aided Geometric Design  
Surface constructions of polynomial degree (3,3) come in four flavours that complement each other: one pair extends the subdivision paradigm, the other the NURBS patch approach to free-form modeling.  ...  facets join, Polar subdivision nicely models regions where many facets join as when capping extruded features.  ...  for polar layout because it generates a finite number of bi-cubic patches in the transition from a quad to a triangle facet while Quad-tri subdivision creates infinitely many patches, half of them three-sided  ... 
doi:10.1016/j.cagd.2008.06.002 fatcat:s45bxcyjefbghjpeiga5fwlgyq

A multisided C2 B-spline patch over extraordinary vertices in quadrilateral meshes

Gerben J. Hettinga, Jiří Kosinka
2020 Computer-Aided Design  
A multisided C-2 B-spline patch over extraordinary vertices in quadrilateral meshes Hettinga, Gerben J.; Kosinka, Jiří  ...  Acknowledgement We gratefully acknowledge the support of NVIDIA Corporation with the donation of the Titan V GPU used for this research.  ...  This process extends standard uniform B-spline subdivision masks/stencils also to faces with any number of sides (extraordinary faces) and vertices with valency other than four (extraordinary vertices)  ... 
doi:10.1016/j.cad.2020.102855 fatcat:7g6dflkklzditm7zpgehe4syym

Surfaces with polar structure

K. Karčiauskas, J. Peters
2007 Computing  
Polar layout is particularly suitable for high valences and is, for example, generated by a new class of subdivision schemes.  ...  We describe the structure and general properties of surfaces with polar layout.  ...  Five-sided regions or, equivalently, new extraordinary points of valence five allow transition from standard subdivision to polar subdivision (Figure 8, left) as follows.  ... 
doi:10.1007/s00607-006-0207-x fatcat:ocyqqv3snrcbtlyi5kaz6zdocu

Refinable C1 spline elements for irregular quad layout

Thien Nguyen, Jörg Peters
2016 Computer Aided Geometric Design  
Building on a result of U. Reif on removable singularities, we construct C1 bi-3 splines that may include irregular points where less or more than four tensor-product patches meet.  ...  As in the regular case, each quadrilateral has four degrees of freedom, each associated with one spline and the splines are linearly independent.  ...  Isogeometric analysis with subdivision surfaces, 2013.  ... 
doi:10.1016/j.cagd.2016.02.009 pmid:27103754 pmcid:PMC4834718 fatcat:lrk2xnh2ajfxnles4gjcgoeeaq

Smoothing Polyhedra Using Trimmed Bicubic Patches [chapter]

J. Peters
1998 Geometric Modelling  
Explicit formulas relating the coefficients of the patches to the vertices of an arbitrary input polyhedron are given. Four of these patches can be grouped together into a NURBS surface.  ...  Several efficient constructions of smooth surfaces following the outlines of a polyhedral mesh are based on three-sided patches.  ...  To show that three-sided patches fit rather nicely into the B-spline standard, the author has worked out the details of representing the three-sided surface splines defined in [3] as linearly-trimmed bicubic  ... 
doi:10.1007/978-3-7091-6444-0_16 dblp:conf/gm/Peters96 fatcat:34c2x7uwxberflz5lppipoa3jm

Assessment of an isogeometric approach with Catmull–Clark subdivision surfaces using the Laplace–Beltrami problems

Zhaowei Liu, Andrew McBride, Prashant Saxena, Paul Steinmann
2020 Computational Mechanics  
The performance of the Catmull-Clark subdivision method is compared to the conventional finite element method. Subdivision surfaces without extraordinary vertices show the optimal convergence rate.  ...  The scalar-valued Laplace-Beltrami equation requires only C 0 continuity and is adopted to elucidate key features and properties of the isogeometric method using Catmull-Clark subdivision surfaces.  ...  subdivision element with an extraordinary vertex. b One level of subdivision of the element patch divides the element into four sub-elements; three of them are in regular patches. c Successive subdivisions  ... 
doi:10.1007/s00466-020-01877-3 fatcat:q4m3e27z2ncgtjtohsyo2rwkvi

Modeling surfaces of arbitrary topology using manifolds

Cindy M. Grimm, John F. Hughes
1995 Proceedings of the 22nd annual conference on Computer graphics and interactive techniques - SIGGRAPH '95  
This is achieved by modeling the domain of the surface with a manifold whose topology matches that of the polyhedral mesh, then embedding this domain into 3-space using a basis-function/control-point formulation  ...  The technique inherits many of the properties of B-splines: local control, a compact representation, and guaranteed continuity of arbitrary degree.  ...  Left: the four regions UV F i Right: the four regions UV E i . An edge chart. Left: the two regions UEF i (F0 is 3-sided, F1 is 4-sided).  ... 
doi:10.1145/218380.218475 dblp:conf/siggraph/GrimmH95 fatcat:owxwqr33rrabfcigw5odlyeqsy

A unified approach to subdivision algorithms near extraordinary vertices

Ulrich Reif
1995 Computer Aided Geometric Design  
We present a unified approach to subdivision algorithms for meshes with arbitrary topology which admits a rigorous analysis of the generated surface and give a sufficient condition for the regularity of  ...  The criterion is easily applicable to all known algorithms such as those of Doo-Sabin and Catmull-Clark, but will also be useful to construct new algorithms like interpolatory subdivision schemes.  ...  Let X be a parametrically smooth spline surface x consisting of rectangular patches xJ,j c J, with a single n-sided hole.  ... 
doi:10.1016/0167-8396(94)00007-f fatcat:5z2xfx4lijcivj2kuseey7t7zm

Page 4719 of Mathematical Reviews Vol. , Issue 94h [page]

1994 Mathematical Reviews  
of which may if desired be chopped into four three- sided quadratic Bézier patches) followed by further work to de- termine the Bézier coefficients of each cubic patch.  ...  The resulting surface is C', and is made up predominantly of four-sided biquadratic patches—similar to a tensor-product biquadratic spline surface—but with small regions (corresponding to the exceptional  ... 

An Introduction to Guided and Polar Surfacing [chapter]

Jörg Peters, Kȩstutis Karčiauskas
2010 Lecture Notes in Computer Science  
with industry standards, either as a finite number of polynomial patches or as a subdivision process.  ...  An example construction generating guided C 2 surfaces from patches of degree bi-3 highlights the power of the approach.  ...  In either case the multi-sided holes have no corresponding box-spline basis function but are filled, either with a cap of n (macro-)patches or via subdivision, i.e. by an infinite sequence of surface rings  ... 
doi:10.1007/978-3-642-11620-9_20 fatcat:d7ou4xj4sbhoraouioiw2sagdi

Extending Catmull-Clark Subdivision and PCCM with Polar Structures

Ashish Myles, Kestutis Karciauskas, Jorg Peters
2007 15th Pacific Conference on Computer Graphics and Applications (PG'07)  
We show that a single NURBS patch can be used where polar subdivision would be applied.  ...  Together, they form a powerful combination for smooth object design: while Catmull-Clark subdivision is more suitable where few facets join, polar subdivision nicely models regions where many facets join  ...  We start by showing that bi-cubic spline subdivision of the NURBS patch does not yield a useful analysis. The radial subdivision matrix of the NURBS patch combines n curve subdivision matrices.  ... 
doi:10.1109/pg.2007.11 dblp:conf/pg/MylesKP07 fatcat:g2v35esu7vgtzmox445vp7jjdm
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