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Approximation and geometric modeling with simplex B-splines associated with irregular triangles

S. Auerbach, R.H.J. Gmelig Meyling, M. Neamtu, H. Schaeben
1991 Computer Aided Geometric Design  
Approximation and geometric modeling with simplex B-splines associated with irregular triangles, Computer Aided Geometric Design 8 (1991) 67-87.  ...  Thus, bivariate quadratic B-splines associated with irregular triangles provide a great flexibility lo approximate and model fast changing or even functions with any given discontinuities from scattered  ...  Traas, Twente, for many instructive discussions on splines and their practical applicability.  ... 
doi:10.1016/0167-8396(91)90050-l fatcat:gbafi4wvenh3relzkmlwmk4a2e

Page 6150 of Mathematical Reviews Vol. , Issue 91K [page]

1991 Mathematical Reviews  
(D-BONN) Approximation and geometric modeling with simplex B-splines associated with irregular triangles. Comput. Aided Geom. Design 8 (1991), no. 1, 67-87.  ...  Thus, bivariate quadratic B- splines associated with irregular triangles provide a great flexibility to approximate and model fast changing functions, or even func- tions with any given discontinuities  ... 

Triangular NURBS and their dynamic generalizations

Hong Qin, Demetri Terzopoulos
1997 Computer Aided Geometric Design  
To enhance the power of this model, we propose triangular NURBS, the rational generalization of triangular B-splines, with weights as additional degrees of freedom.  ...  Triangular B-splines are a new tool for modeling a broad class of objects defined over arbitrary, nonrectangular domains.  ...  Acknowledgments We would like to thank Professor Hans-Peter Seidel of Universit~it Erlangen for kindly making available the software for triangular B-spline surfaces that he developed with Mr.  ... 
doi:10.1016/s0167-8396(96)00062-3 fatcat:5pyhbzjurzh2hggwdeaai4dro4

Splines and geometric modeling [chapter]

Chandrajit Bajaj
2004 Handbook of Discrete and Computational Geometry, Second Edition  
Neamtu, and H. Schaeben. Approximation and geometric modeling with simplex B splines associated with irregular triangles. Computer Aided G e ometric Design, 8:67 87, 1991. Baj92 C. Bajaj.  ...  These multivariate splines are de ned as projections of simplices and are therefore called simplex splines. Auerbach AMNS91 constructs approximations with simplex splines over irregular triangles.  ... 
doi:10.1201/9781420035315.ch53 fatcat:e7evq2b77ba5lj3vxlcxwxxzda

Dynamic manipulation of triangular B-splines

Hong Qin, Demetri Terzopoulos
1995 Proceedings of the third ACM symposium on Solid modeling and applications - SMA '95  
We demonstrate several applications of dynamic triangular B-splines, including interactive sculpting using forces and physical parameters, the fitting of unstructured data, and solid rounding with geometric  ...  Dynamic triangular B-splines provide a systematic and unified approach for a variety of solid modeling problems including shape blending, constraint-based design, and parametric design.  ...  Acknowledgments We are grateful to Professor Hans-Peter Seidel for kindly making available the software for triangular B-spline surfaces that he developed with Philip Fong.  ... 
doi:10.1145/218013.218084 dblp:conf/sma/QinT95 fatcat:aadkvyhr7rf2zhnzruekddjqny

Construction of C^2 cubic splines on arbitrary triangulations [article]

Tom Lyche, Carla Manni, Hendrik Speleers
2021 arXiv   pre-print
We provide a simplex spline basis for the space of C^2 cubics defined on a single macro-triangle which behaves like a Bernstein/B-spline basis over the triangle.  ...  The C^2 cubic space on such a refined triangulation has a stable dimension and optimal approximation power.  ...  Domain points and condition number We now associate a special point in ∆ with each basis function B i in (4), that plays an important role in geometric modeling.  ... 
arXiv:2110.07907v1 fatcat:i5qlrbl6czg5fgt3vx36rppusq

Physically based modeling and simulation with dynamic spherical volumetric simplex splines

Yunhao Tan, Jing Hua, Hong Qin
2010 Computer-Aided Design  
In this paper, we present a novel computational modeling and simulation framework based on dynamic spherical volumetric simplex splines.  ...  In this framework, we first develop an accurate and efficient algorithm to reconstruct the high-fidelity digital model of a real-world object with spherical volumetric simplex splines which can represent  ...  grants MTTC05-135/GR686 and MTTC05-154/GR705, and the Michigan 21st Century Jobs Funds 06-1-P1-0193.  ... 
doi:10.1016/j.cad.2009.02.014 pmid:20161636 pmcid:PMC2817959 fatcat:lcc443ax3rd2bktiywrgtfj56a

Multiresolution Heterogeneous Solid Modeling and Visualization Using Trivariate Simplex Splines [article]

J. Hua, Y. He, H. Qin
2004 Solid Modeling  
Heterogeneous material attributes associated with solid geometry can be modeled and edited by manipulating the control vectors and/or associated knots of trivariate simplex splines easily.  ...  In addition, based on the simplex spline theory, we derive several theoretical formula and propose a fast direct rendering algorithm for interactive data analysis and visualization of the simplex spline  ...  Nonetheless, modeling with B-splines or NURBS has severe shortcomings. Its modeling scope is extremely constrained in term of geometric, topological, and attribute aspects.  ... 
doi:10.2312/sm.20041376 fatcat:n5wkunhci5cdtltyfziyzri4g4

3D modeling of natural heterogeneous objects: an introduction to methods and algorithms

Miller Gomez-Mora, Rocio Rodriguez-Guerrero
2018 Contemporary Engineerng Sciences  
This document provides a general description of the most well-known modeling methods and algorithms with a discussion of relative advantages and disadvantages for using them in generation of high-fidelity  ...  The field of three-dimensional (3D) modeling of natural heterogeneous object is relatively new and its results have many applications in science and engineering.  ...  NURBS can be used as a method for approximation or interpolation of scattered data and is also incorporated with most of the current geometric modeling systems for reconstructing smooth surfaces and dealing  ... 
doi:10.12988/ces.2018.88459 fatcat:rz3wikdyyzgs5alrsvphy5zqce

PHYSICS BASED GEOMETRIC DESIGN

HONG QIN
1996 International journal of shape modeling  
Geometric modeling has proved to be crucial to computer graphics and computer aided geometric design (CAGD).  ...  In this paper, we s u r v ey the diversity of shape representations ranging from the primitive polynomial to the sophisticated Non-Uniform Rational B-Spline (NURBS).  ...  Figure 5 : 5 An irregular surface and its parametric domain. carried over to trans nite interpolation of triangles 5]. tion theory.  ... 
doi:10.1142/s0218654396000099 fatcat:ctnoubb7vvaubpbxuiev5dpmpe

Generalized B-spline surfaces of arbitrary topology

Charles Loop, T. D. DeRose
1990 Computer graphics  
It is not possible to model a general closed surface or a surface with handles as a single non-degenerate B-spline. In practice such surfaces are often needed.  ...  In this paper, we present generalizations of biquadratic and bicubic B-spline surfaces that are capable of capturing surfaces of arbitrary topology (although restrictions are placed on the connectivity  ...  However, the current theory of B-splines has serious shortcomings when modeling general closed surfaces or surfaces with handles.  ... 
doi:10.1145/97880.97917 fatcat:2lawxbwjabevhhzrkwnf5vxjr4

Corner-Sharing Tetrahedra for Modeling Micro-Structure [article]

Meera Sitharam, Jeremy Youngquist, Maxwell Nolan, Jörg Peters
2018 arXiv   pre-print
We propose modeling micro-structure with a class of discrete Corner-Sharing Tetrahedra (CoSTs).  ...  The paper develops a palette of simple geometry processing operations on CoSTs including graph manipulation, hierarchical refinement, randomization, and generating associated continuous representations  ...  (b) 3-direction box-spline level set. Figure 15 : 15 Parametric spline associated with bivariate CoST layers fromFig. 3. Each spline patch is shown in a different color.  ... 
arXiv:1806.05528v1 fatcat:a5arrposebdilpoubdul3ju3te

Generalized B-spline surfaces of arbitrary topology

Charles Loop, T. D. DeRose
1990 Proceedings of the 17th annual conference on Computer graphics and interactive techniques - SIGGRAPH '90  
It is not possible to model a general closed surface or a surface with handles as a single non-degenerate B-spline. In practice such surfaces are often needed.  ...  In this paper, we present generalizations of biquadratic and bicubic B-spline surfaces that are capable of capturing surfaces of arbitrary topology (although restrictions are placed on the connectivity  ...  However, the current theory of B-splines has serious shortcomings when modeling general closed surfaces or surfaces with handles.  ... 
doi:10.1145/97879.97917 dblp:conf/siggraph/LoopD90 fatcat:dqcqycgm3bfidfqvkvop6lqbni

Incorporating Rigid Structures in Non-rigid Registration Using Triangular B-Splines [chapter]

Kexiang Wang, Ying He, Hong Qin
2005 Lecture Notes in Computer Science  
In contrast to the conventional registration method based on tensor-product B-splines, our approach recovers local rigid transformation with fewer degrees of freedom (DOFs), and accurately simulates sharp  ...  control and space-varying smoothness modeling.  ...  Acknowledgements This research was partially supported by the NSF grant ACI-0328930, the ITR grant IIS-0326388, and the Alfred P. Sloan Fellowship.  ... 
doi:10.1007/11567646_20 fatcat:lbcoo4ezbzeilcft567zjzgb7i

Edge subdivision schemes and the construction of smooth vector fields

Ke Wang, Weiwei, Yiying Tong, Mathieu Desbrun, Peter Schröder
2006 ACM Transactions on Graphics  
Vertex-and face-based subdivision schemes are now routinely used in geometric modeling and computational science, and their primal/dual relationships are well studied.  ...  Using Loop's scheme for 0-forms and generalized half-box splines for 2forms results in a unique generalized spline scheme for 1-forms, easily incorporated into standard subdivision surface codes.  ...  Information, nVidia, and Autodesk.  ... 
doi:10.1145/1141911.1141991 fatcat:guhqrub2tbgrbll2gbvwk6vgey
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