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B-spline free-form deformation of polygonal object as trimmed Bézier surfaces

2002
*
The Visual Computer
*

We also thank the reviewers who gave helpful suggestions

doi:10.1007/s00371-002-0171-1
fatcat:e7ajdrarerdjdbzdjbjm6gezha
*on*improving the quality of the paper. ... Polynomial interpolation for*tensor*-*product**Bézier**surface*First, the sampling points are computed*on*the*surface*. ... Finally the*tensor*-*product**Bézier**surface*is evaluated through polynomial interpolation, and the deformation results in the trimmed*tensor*-*product**Bézier**surfaces*. ...##
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Drawing Rational Surfaces with Rational Bezier Triangles

2007
*
Interdisciplinary Information Sciences
*

We also give a method for drawing a rational

doi:10.4036/iis.2007.239
fatcat:fzlbgzzl4bct5dpcopwe43bk5u
*Bézier**triangle*with a rational*tensor**product**Bézier**surface*. ... This paper deals with a new conception "dual*Bézier**triangle*" and we develop a method for drawing a rational*tensor**product**Bézier**surface*with two rational*Bézier**triangles*. ... 1 ,*on*the other hand, the rational*tensor**product**Bézier**surfaces*are defined over D, so Theorem 4.2 is not suitable to draw a rational*Bézier**triangle*with a rational*tensor**product**Bézier**surface*. ðvÞ ...##
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A multisided generalization of Bézier surfaces

1989
*
ACM Transactions on Graphics
*

In this paper we introduce a class of

doi:10.1145/77055.77059
fatcat:gmascpgiajactjoagm7urcsu5q
*surface*patch representations, called S-patches, that unify and generalize triangular and*tensor**product**Bezier**surfaces*by allowing patches to be defined over any ... Of special interest are the regular S-patches, that is, S-patches defined*on*regular domain polygons. Also presented is an algorithm for smoothly joining together these*surfaces*with C" continuity. ... Interpretation 3 4*Triangle*+*Tensor**product*4 3*Tensor**product*+*Triangle*3 n*Triangle*-S-patch 4*Tensor**product*+ S-patch m ; S-patch -+*Triangle*m 4 S-patch -*Tensor**product*PROOF. ...##
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A geometric interpretation of the diagonal of a tensor-product Bézier volume

1999
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Computer Aided Geometric Design
*

A geometric interpretation of the diagonal of a

doi:10.1016/s0167-8396(99)00004-7
fatcat:kyg6ciecnfakzamqjangrjnb74
*tensor*-*product*trivariate*Bézier*volume using degree elevation of*Bézier**triangles*is given. ... Diagonal of a*tensor*-*product**Bézier*volume Now suppose we have a*tensor*-*product*trivariate*Bézier*volume given by x(u, v, w) = m i=0 n j =0 p k=0 b ij k B m i (u)B n j (v)B p k (w). ... For a*tensor*-*product**Bézier*volume of degrees m, n, and p the diagonal curve will have degree m + n + p. We will focus*on*the case where m = n = p and the degree of the diagonal curve is 3n. ...##
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SLIME: A new deformable surface

1994
*
Procedings of the British Machine Vision Conference 1994
*

The method is based

doi:10.5244/c.8.28
dblp:conf/bmvc/StoddartHI94
fatcat:mtauy66xd5dtbd52hkebn2pswi
*on*a generalization of biquadratic B-splines, and has a comparable computational cost to methods based*on*traditional*tensor**product*B-splines. ... This means that the*surface*topology is fixed in advance, and also imposes severe limitations*on*how a*surface*can be described. ... It both unifies and generalizes the*Bezier**triangle*and the*tensor**product*4-sided*Bezier*patch. ...##
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Real-time GPU rendering of piecewise algebraic surfaces

2006
*
ACM Transactions on Graphics
*

Figure 1: Several states of an animated fourth order algebraic

doi:10.1145/1141911.1141939
fatcat:t5yqloktxne2zlou2aq2gnl5ea
*surface*rendered in real-time using our technique. ... Using*tensors*allows us to express and evaluate blossoms in terms of dot*products*that are native operations*on*a GPU. ... The mechanism we use is to store a variant of the*tensor*B as attributes*on*the vertices of*triangles*. The GPU will interpolate vertex attributes when rasterizing a*triangle*. ...##
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Real-time GPU rendering of piecewise algebraic surfaces

2006
*
ACM SIGGRAPH 2006 Papers on - SIGGRAPH '06
*

Abstract We consider the problem of real-time GPU rendering of algebraic

doi:10.1145/1179352.1141939
fatcat:hkiv4opwlrbhvf5brff3mzfl4u
*surfaces*defined by*Bézier*tetrahedra. ... These*surfaces*are rendered directly in terms of their polynomial representations, as opposed to a collection of approximating*triangles*, thereby eliminating tessellation artifacts and reducing memory ... Using*tensors*allows us to express and evaluate blossoms in terms of dot*products*that are native operations*on*a GPU. ...##
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Composition of Tensor Product Bézier Representations
[chapter]

1993
*
Geometric Modelling
*

This paper shows how to compose polynomial and rational

doi:10.1007/978-3-7091-6916-2_11
fatcat:fapycxobkbbpflsntz5rxrowsm
*tensor**product**Bezier*representations. ... Trimming of*surfaces*and volumes, curve and*surface*modeling via Bezier's idea of destortion, segmentation, reparametrization, geometric continuity are examples of applications of functional composition ... A*tensor**product**Bezier**surface*-briefl.y TPB-*surface*-of degree (1, m) is defined by l m X(u,v) L L bi.J BJ(u) Bj(v), u,v E [0,1), i=Oj=O and a*tensor**product**Bezier*volume -briefl.y TPB-volume ...##
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Real-time accurate free-form deformation in terms of triangular Bézier surfaces

2014
*
Applied Mathematics-A Journal of Chinese Universities
*

*Bézier*

*surfaces*, which are fully implemented via CUBLAS

*on*GPU. • Adopting of vertex buffer objects for OpenGL rendering to reduce data transfer. ... But it contains intensive computations, such as B-spline volume evaluations,

*Bézier*

*surface*interpolations,

*Bézier*

*surface*tessellations, etc. ... First, adaptive tessellation can save graphics memory, since it can prevent the generation of too many tiny

*triangles*or flat patches. However, it is complex to be implemented

*on*GPU. ...

##
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On the CAD-compatible conversion of S-patches
[article]

2020
*
arXiv
*
pre-print

It is known since their first appearance, that any regular S-patch can be exactly converted into a trimmed rational B\'ezier

arXiv:2002.11111v1
fatcat:jww3kyljbbh3zfs3aqapet36r4
*surface*. ... This is a big advantage compared to other multi-sided*surface*representations that have to be approximated for exporting them into CAD/CAM systems. ... First the n-sided S-patch is replaced with a four-sided*one*(Section 4.1), then the*tensor**product*form is computed (Section 4.2), resulting in a*surface*of degree (n − 2)d. ...##
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Refinement for a Hybrid Boundary Representation and its Hybrid Volume Completion

2019
*
SMAI Journal of Computational Mathematics
*

Standard refinement approches to

doi:10.5802/smai-jcm.49
fatcat:2etaujciovehtlg7bjgpdzytfa
*tensor**product*B-splines and subdivision of*Bézier**triangles*and tetrahedra must be adapted to this representation. ... The revised and trim basis functions for HB-and HV-rep depend*on*a partition of knot intervals. Therefore, a naive h-refinement can change basis functions in unexpected ways. ... The*Bézier*pyramid [4] has been shown to have the boundary geometry of*Bézier**triangles**on*each of the 4 faces, and a*tensor**product*degree bi-d*Bézier**on*its base*surface*. ...##
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Hybrid volume completion with higher-order Bézier elements

2015
*
Computer Aided Geometric Design
*

The

doi:10.1016/j.cagd.2015.03.008
fatcat:bjoklf5tnrf7vlqfwswjdkzaay
*product*consists of trivariate*tensor**product*B-Splines near the boundary and unstructured higher-order*Bézier*tetrahedral elements in the interior of the object with C 0 smoothness across their interfaces ... Then, a intermediate interface consisting of*Bézier**triangles*is built to match the inner boundary of the B-Spline representation. ...*surface*, a*tensor**product**Bézier**surface*, and a*Bézier**triangle**surface*. ...##
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Approximate implicitization of triangular Bézier surfaces

2010
*
Proceedings of the 26th Spring Conference on Computer Graphics - SCCG '10
*

We discuss how Dokken's methods of approximate implicitization can be applied to triangular

doi:10.1145/1925059.1925084
dblp:conf/sccg/BarrowcloughD10
fatcat:r5zo6qxqmzc2fngdhc6wxu42tm
*Bézier**surfaces*in both the original and weak forms. ... We would like to thank Johan Simon Seland at SINTEF for making his real-time algebraic*surface*visualization system available to us (Figure 1) . ... While approximate implicitization of*tensor*-*product**Bézier**surfaces*is a fairly simple extension of approximate implicitization of 2D rational parametric curves, triangular*Bézier**surfaces*are somewhat ...##
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Tensor product surfaces as rewriting process

2006
*
Proceedings of the 22nd Spring Conference on Computer Graphics - SCCG '06
*

We demonstrate their functionality

doi:10.1145/2602161.2602173
dblp:conf/sccg/KolingerovaMB06
fatcat:awdodlpq35haragxzad4eoh3aq
*on**Bézier*bicubic and rational*Bézier*bicubic*surfaces*. ... We propose using L-systems for*tensor**product**surfaces*subdivision description. The parallel rewriting helps us to merge the parallel manner of a*surface*subdivision. ... The*Bézier*bicubic*tensor**product**surface*is given by the sixteen control vertices v 0,0 ,... ...##
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Bézier Representation of Trim Curves
[chapter]

1995
*
Geometric Modelling
*

The composition of

doi:10.1007/978-3-7091-7584-2_16
fatcat:yl4gwczvrbenbgypi2r2gz6r6i
*Bezier*curves and*tensor**product**Bezier**surfaces*, polynomial as weil as rational, is applied to exactly and explicitely represent trim curves of*tensor**product**Bezier**surfaces*. ... A*Bezier*spline approximation of lower polynomial degree is built up as weil which is based*on*the ' exact trim curve representation in coordinate space. ... polynomial*tensor**product**Bezier**surface*trimmed by polynomial*Bezier*curves into a composite*Bezier**surface*. ...
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