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

Jieqing Feng, Tomoyuki Nishita, Xiaogang Jin, Qunsheng Peng
2002 The Visual Computer
We also thank the reviewers who gave helpful suggestions 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.  ...

### Drawing Rational Surfaces with Rational Bezier Triangles

Masato FUKUSHIMA
2007 Interdisciplinary Information Sciences
We also give a method for drawing a rational 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  ...

### A multisided generalization of Bézier surfaces

Charles T. Loop, Tony D. DeRose
1989 ACM Transactions on Graphics
In this paper we introduce a class of 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.  ...

### A geometric interpretation of the diagonal of a tensor-product Bézier volume

D.J. Holliday, G.E. Farin
1999 Computer Aided Geometric Design
A geometric interpretation of the diagonal of a 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.  ...

### SLIME: A new deformable surface

A. J. Stoddart, A. Hilton, J. Illingworth
1994 Procedings of the British Machine Vision Conference 1994
The method is based 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.  ...

### Real-time GPU rendering of piecewise algebraic surfaces

Charles Loop, Jim Blinn
2006 ACM Transactions on Graphics
Figure 1: Several states of an animated fourth order algebraic 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.  ...

### Real-time GPU rendering of piecewise algebraic surfaces

Charles Loop, Jim Blinn
2006 ACM SIGGRAPH 2006 Papers on - SIGGRAPH '06
Abstract We consider the problem of real-time GPU rendering of algebraic 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.  ...

### Composition of Tensor Product Bézier Representations [chapter]

D. Lasser
1993 Geometric Modelling
This paper shows how to compose polynomial and rational 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  ...

### Real-time accurate free-form deformation in terms of triangular Bézier surfaces

Yuan-min Cui, Jie-qing Feng
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.  ...

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

Péter Salvi
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 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.  ...

### Refinement for a Hybrid Boundary Representation and its Hybrid Volume Completion

Yang Song, Elaine Cohen
2019 SMAI Journal of Computational Mathematics
Standard refinement approches to 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.  ...

### Hybrid volume completion with higher-order Bézier elements

Suqin Zeng, Elaine Cohen
2015 Computer Aided Geometric Design
The 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.  ...

### Approximate implicitization of triangular Bézier surfaces

Oliver J. D. Barrowclough, Tor Dokken
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 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  ...

### Tensor product surfaces as rewriting process

Ivana Kolingerová, Petr März, Bedrich Beneš
2006 Proceedings of the 22nd Spring Conference on Computer Graphics - SCCG '06
We demonstrate their functionality 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 ,...  ...

### Bézier Representation of Trim Curves [chapter]

D. Lasser, G.-P. Bonneau
1995 Geometric Modelling
The composition of 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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