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Explicit matrix representation for NURBS curves and surfaces

Ligang Liu, Guojin Wang
2002 Computer Aided Geometric Design  
In this paper we have presented two matrix representation formulations for arbitrary degree NURBS curves and surfaces explicitly other than recursively.  ...  The matrix forms for curves and surfaces were largely promoted in CAD/CAM.  ...  Acknowledgements We would like to thank the anonymous reviewers for their comments.  ... 
doi:10.1016/s0167-8396(02)00124-3 fatcat:xemy722ogfgtfgqboxfgfdgbmi

A General Class of C1 Smooth Rational Splines: Application to Construction of Exact Ellipses and Ellipsoids

Hendrik Speleers, Deepesh Toshniwal
2020 Computer-Aided Design  
In this paper, we describe a general class of C 1 smooth rational splines that enables, in particular, exact descriptions of ellipses and ellipsoids -some of the most important primitives for CAD and CAE  ...  To this end, we explicitly present the matrices (a) that describe our splines in terms of NURBS, and (b) that help refine the splines by performing (local) degree elevation and knot insertion.  ...  Speleers was partially supported by the Beyond Borders Program of the University of Rome Tor Vergata, Italy through the project ASTRID (CUP E84I19002250005) and by the MIUR Excellence Department Project  ... 
doi:10.1016/j.cad.2020.102982 fatcat:jeptshga6bgqnd4robn4n7rmfq

Dynamic NURBS with geometric constraints for interactive sculpting

Demetri Terzopoulos, Hong Qin
1994 ACM Transactions on Graphics  
We also introduce a new technique for 2D shape metamorphosis using constrained D-NURBS surfaces. . 105 1Note, however, that, for static weights, the matrices become time invariant and the computational  ...  To derive these equations, we employ Lagrangian mechanics and a finite-element-like discretization. Our approach supports the trimming of D-NURBS surfaces using D-NURBS curves.  ...  Curves For simplicity, consider first a D-NURBS space curve.  ... 
doi:10.1145/176579.176580 fatcat:ttoickulhjgz3oaxwb4f5wvutq

Page 4587 of Mathematical Reviews Vol. , Issue 2003f [page]

2003 Mathematical Reviews  
(PRC-ZHJ-KL; Hangzhou) ; Wang, Guojin [Wang, Guo Jin'| (PRC-ZHJ-KL: Hangzhou) Explicit matrix representation for NURBS curves and surfaces.  ...  The main result is an explicit formula for the coefficients of NURBS based on computation of divided difference of a con- tinuous real function.  ... 

Page 7516 of Mathematical Reviews Vol. , Issue 2002J [page]

2002 Mathematical Reviews  
Summary: “An algorithmic approach to degree elevation of NURBS curves is presented. The new algorithms are based on the weighted blossoming process and its matrix representation.  ...  {For the entire collection see MR 2002g:65005. } 2002j:65026 65D17 Koua Brou, J. C. Explicit representation for a C' piecewise rational surfaces defined over a triangulation.  ... 

D-NURBS: a physics-based framework for geometric design

Hong Qin, D. Terzopoulos
1996 IEEE Transactions on Visualization and Computer Graphics  
We use Lagrangian mechanics to formulate the equations of motion for D-NURBS curves, tensor-product D-NURBS surfaces, swung D-NURBS surfaces, and triangular D-NURBS surfaces.  ...  We implement a p r ototype modeling environment based on D-NURBS, and demonstrate that D-NURBS can be e ective tools in a wide range of CAGD applications such as shape blending, scattered data tting, and  ...  Acknowledgements We w ould like to thank Professor Hans-Peter Seidel for kindly making available the software for triangular Bspline surfaces that he developed with Philip Fong.  ... 
doi:10.1109/2945.489389 fatcat:iaea3xyf6jfb5bevwaw6pve4bq

Page 4321 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews  
The representation is in the form of a cubic Bernstein- Bézier triangle and is invariant under affine transformations of the curve.  ...  Summary: “An explicit expression for the unique implicitization of any planar rational cubic Bézier curve is presented, the only exception being when the tangents at the ends of the curve are parallel.  ... 

A NURBS enhanced extended finite element approach for unfitted CAD analysis

Grégory Legrain
2013 Computational Mechanics  
A NURBS Enhanced eXtended Finite Element Approach is proposed for the unfitted simulation of structures defined by means of CAD parametric surfaces.  ...  Finally, a geometrically implicit/explicit approach is proposed for efficiency purpose in the context of fracture mechanics.  ...  It was also proposed to use an implicit/explicit representation of the geometry: explicit for the general geometry of the structure and implicit for evolving features such as cracks.  ... 
doi:10.1007/s00466-013-0854-7 fatcat:npdrdtw4brfd7kyzq33kv6o7x4

Efficient 6DOF tools for free-form surface modelling

Alfredo Liverani, Serena Morigi
2004 The Visual Computer  
PS lets the designer generate a new surface by interactively and automatically adding section curves to a non-uniform rational B-spline surface.  ...  The proposed new tools are progressive skinning (PS) and curve-over-surface shaping (COSS), both implemented on a bimanual virtual environment.  ...  As for many NURBS algorithms, we simply operate on the four coordinates homogeneous representation of rational curves and surfaces.  ... 
doi:10.1007/s00371-004-0258-y fatcat:fzgfobnhufg5pllb3mp4ya3c4i

Parametric reconstruction of generalized cylinders from limb edges

Chunhong Pan, Hongping Yan, G. Medioni, Songde Ma
2005 IEEE Transactions on Image Processing  
Then, for different types of GCs, three kinds of symmetries (parallel symmetry, skew symmetry, and local smooth symmetry) are employed to compute the symmetry of limb edges.  ...  The surface points corresponding to limb edges in images are reconstructed by integrating the recovered epipolar geometry and the properties induced from the assumptions that we make on the GCs.  ...  ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their thorough review of the paper and many constructive comments.  ... 
doi:10.1109/tip.2005.847296 pmid:16121466 fatcat:n2a4qbwtczciljhi2dlmejh274

Optimized stereo reconstruction of free-form space curves based on a nonuniform rational B-spline model

Yi Jun Xiao, Y. F. Li
2005 Optical Society of America. Journal A: Optics, Image Science, and Vision  
weights of a NURBS representation of the curve, accordingly bypassing the error-prone point-to-point correspondence matching.  ...  We present an optimization framework for such a problem based on a nonuniform rational B-spline (NURBS) curve model that converts reconstruction of a 3D curve into reconstruction of control points and  ...  Wahl, and the reviewers for their useful comments and help with the research.  ... 
doi:10.1364/josaa.22.001746 pmid:16211801 fatcat:svomrjuuyjahvms3ef4sdhcyvi

Full analytical sensitivities in NURBS based isogeometric shape optimization

Xiaoping Qian
2010 Computer Methods in Applied Mechanics and Engineering  
Non-uniform rational B-spline (NURBS) has been widely used as an effective shape parameterization technique for structural optimization due to its compact and powerful shape representation capability and  ...  The analytical sensitivity for both positions and weights of NURBS control points is especially beneficial for recovering optimal shapes that are conical e.g. ellipses and circles in 2D, cylinders, ellipsoids  ...  Note, without weights, NURBS shape degenerates into B-spline shape and B-spline representation cannot exactly represent these conic curves and surfaces.  ... 
doi:10.1016/j.cma.2010.03.005 fatcat:2wdamekclrgyhkmadtjpvx5vqi

GPU-based trimming and tessellation of NURBS and T-Spline surfaces

Michael Guthe, Aákos Balázs, Reinhard Klein
2005 ACM SIGGRAPH 2005 Papers on - SIGGRAPH '05  
Abstract As there is no hardware support neither for rendering trimmed NURBS -the standard surface representation in CAD -nor for T-Spline surfaces the usability of existing rendering APIs like OpenGL,  ...  Combining this new method with GPU-based tessellation of cubic rational surfaces allows a new rendering algorithm for arbitrary trimmed NURBS and T-Spline surfaces with prescribed error in screen space  ...  reduce p-buffer switches when rendering multiple trimmed surfaces, and Andreas Schilling for his valuable comments.  ... 
doi:10.1145/1186822.1073305 fatcat:holhtj73j5gj7cfbhmb2e35paa

GPU-based trimming and tessellation of NURBS and T-Spline surfaces

Michael Guthe, Ákos Balázs, Reinhard Klein
2005 ACM SIGGRAPH 2005 Sketches on - SIGGRAPH '05  
Abstract As there is no hardware support neither for rendering trimmed NURBS -the standard surface representation in CAD -nor for T-Spline surfaces the usability of existing rendering APIs like OpenGL,  ...  Combining this new method with GPU-based tessellation of cubic rational surfaces allows a new rendering algorithm for arbitrary trimmed NURBS and T-Spline surfaces with prescribed error in screen space  ...  reduce p-buffer switches when rendering multiple trimmed surfaces, and Andreas Schilling for his valuable comments.  ... 
doi:10.1145/1187112.1187190 dblp:conf/siggraph/GutheBK05 fatcat:2ueaqgoub5hcbfwjef6ajaediy

Explicit dynamics in impact simulation using a NURBS contact interface

Peter Otto, Laura Lorenzis, Jörg F. Unger
2019 International Journal for Numerical Methods in Engineering  
Using a spectral element discretization in combination with a NURBS contact layer the bulk part of the mass matrix is diagonal.  ...  For the contact discretization an intermediate non-uniform rational B-spline (NURBS) layer is added between the contacting finite element bodies, which allows a smooth contact formulation and efficient  ...  NURBS curves and surfaces A NURBS curve is given by C( ) = n ∑ i=1 R i ( )x i , (12) where x i , i = 1, …, n are the control points.  ... 
doi:10.1002/nme.6264 fatcat:6sb3n24scbbkxik6xmzrqr275m
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