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Triangular subpatches of rectangular Bézier surfaces
Abstract
A formula is presented for describing triangular subpatches of rectangular Bézier surfaces. Calculations using it are numerically stable, since they are based on de Casteljau recursions and convex combinations of combinatorial constants. Several examples of quadratic, cubic and quartic subpatches are given, and the bi- and the quadripartition of a rectangular Bézier surface are discussed.
... To solve this issue, we propose an approach to represent trimmed NURBS geometries using watertight rTBS patches. Specifically, we directly extract rTBS patches from the The extraction of triangular Bézier subpatches from tensor-product Bézier patches is based on the results in [74]. The composition of a degree n Bézier triangle in the parametric domain and a tensor-product Bézier patch of degree (p, q) can be exactly represented as a Bézier triangle of degree n(p + q). ...
... The composition of a degree n Bézier triangle in the parametric domain and a tensor-product Bézier patch of degree (p, q) can be exactly represented as a Bézier triangle of degree n(p + q). The control points of the resulted triangular Bézier subpatch can also be calculated explicitly [74]. In this paper we only work with n = 1, that is, we use linear triangle in the parametric domain to extract rTBS patches from tensor-product Bézier patches. ...
... ii. According to [74], the composition of tensor-product Bézier patches of degree p, q and Bézier curve of degree n are Bézier curves of degree n(p + q). Although we can compute such high order Bézier curves exactly, to reduce the complexity of the problem, we approximate c 1 , c 2 using piecewise linear segmentsc 1 andc 2 by connecting the knot points. ...
... The extraction of triangular Bézier subpatches from tensor-product Bézier patches is based on the results in [37]. The composition of a degree n Bézier triangle in the parametric domain and a tensor-product Bézier patch of degree (p, q) can be exactly represented as a Bézier triangle of degree n(p + q). ...
... The composition of a degree n Bézier triangle in the parametric domain and a tensor-product Bézier patch of degree (p, q) can be exactly represented as a Bézier triangle of degree n(p + q). The control points of the resulted triangular Bézier subpatch can also be calculated explicitly [37]. In this paper we only work with n = 1, that is, we use linear triangle in the parametric domain to extract rTBS patches from tensor-product Bézier patches. ...
... ii. According to [37], the composition of tensor-product Bézier patches of degree p, q and Bézier curve of degree n are Bézier curves of degree n(p + q). ...
This paper presents an approach for isogeometric analysis of 3D objects using rational Bézier tetrahedral elements. In this approach, both the geometry and the physical field are represented by trivariate splines in Bernstein Bézier form over the tetrahedrangulation of a 3D geometry. Given a NURBS represented geometry, either untrimmed or trimmed, we first convert it to a watertight geometry represented by rational triangular Bézier splines (rTBS). For trimmed geometries, a compatible subdivision scheme is developed to guarantee the watertightness. The rTBS geometry preserves exactly the original NURBS surfaces except for an interface layer between trimmed surfaces where controlled approximation occurs. From the watertight rTBS geometry, a Bézier tetrahedral partition is generated automatically. By imposing continuity constraints on Bézier ordinates of the elements, we obtain a set of global smooth basis functions and use it as the basis for analysis. Numerical examples demonstrate that our method achieve optimal convergence in spaces and can handle complicated geometries.
... However, the algorithm for the control points of the compositions is not sufficiently efficient in practice. Lasser [15, 16] studied the composition of TP over TB, and the composition of TB over TP. Explicit formulae are provided for the control points of the compositions . ...
... Feng and Peng [17] considered a simpler case using shifting operator to derive the composition of a triangle with a TB surface. Lasser [15, 16] formulated the control points of the composition as the linear combinations of some intermediate points called the construction points. However, the number of the construction points is huge and many of them actually have the same positions. ...
This paper considers the problem of computing the Bézier representation for a triangular sub-patch on a triangular Bézier surface. The triangular sub-patch is defined as a composition of the triangular surface and a domain surface that is also a triangular Bézier patch. Based on de Casteljau recursions and shifting operators, previous methods express the control points of the triangular sub-patch as linear combinations of the construction points that are constructed from the control points of the triangular Bézier surface. The construction points contain too many redundancies. This paper derives a simple explicit formula that computes the composite triangular sub-patch in terms of the blossoming points that correspond to distinct construction points and then an efficient algorithm is presented to calculate the control points of the sub-patch.
... The construction of these triangular Bézier patches which are arbitrarily located within the trimmed surface is performed accordingly to Lasser [182]. In general, a Bézier triangle T of degree p in a tensor product basis of a surface R(u, v) of degrees p and q yields a triangular patch S △ (r, s, t) of degree p(p + q). ...
We review the treatment of trimmed geometries in the context of design, data exchange, and computational simulation. Such models are omnipresent in current engineering modeling and play a key role for the integration of design and analysis. The problems induced by trimming are often underestimated due to the conceptional simplicity of the procedure. In this work, several challenges and pitfalls are described.
... The construction of these triangular Bézier patches which are arbitrarily located within the trimmed surface is performed accordingly to Lasser [182]. In general, a Bézier triangle T of degreep in a tensor product basis of a surface R(u, v) of degrees p and q yields a triangular patch S (r, s, t) of degreep (p + q). ...
... Currently, the least squares (LS) method has been most widely used in data fitting. The commonly used basis functions are polynomials [1], rational functions [2], Gaussian, exponential, smoothing spline in curve fitting, the B-spline [3], the nonuniform rational B-splines (NURBS) [4], Bézier surfaces [5], and radial basis function [6] in surface construction. And, simultaneously, the deformations of LS as RLS (recursive least squares), TLS (total least squares), PLS (partial least squares), WLS (weighted least squares), GLS (generalized least squares), and SLS (segmented least squares) have been also put forward. ...
In the electromagnetic field measurement data postprocessing, this paper introduced the moving least squares (MLS) approximation method. The MLS combines the concept of moving window and compact support weighting functions. It can be regarded as a combination of weighted least squares and segmented least square. The MLS not only can acquire higher precision even with low order basis functions, but also has good stability due to its local approximation scheme. An attractive property of MLS is its flexible adjustment ability. Therefore, the data fitting can be easily adjusted by tuning weighting function’s parameters. Numerical examples and measurement data processing reveal its superior performance in curves fitting and surface construction. So the MLS is a promising method for measurement data processing.
We present a method for the generation of higher-order tetrahedral meshes. In contrast to previous methods, the curved tetrahedral elements are guaranteed to be free of degeneracies and inversions while conforming exactly to prescribed piecewise polynomial surfaces, such as domain boundaries or material interfaces. Arbitrary polynomial order is supported. Algorithmically, the polynomial input surfaces are first covered by a single layer of carefully constructed curved elements using a recursive refinement procedure that provably avoids degeneracies and inversions. These tetrahedral elements are designed such that the remaining space is bounded piecewise linearly. In this way, our method effectively reduces the curved meshing problem to the classical problem of linear mesh generation (for the remaining space).
In this thesis we analyse the behaviour of subdivision schemes, tool used to modelise smooth and multi-resolution surfaces. Firstly, our work aimed to study the geometric behaviour of these surfaces on the neighbourhood of the extraordinary vertices in the control mesh. The geometry of a subdivision surface is complex on the neighborhood of an extraordinary vertex, some of the unpleasant behaviours are obscure. We have proposed an evaluation framework for subdivision surfaces throughout an suitable measure : the absolute curvature gradient. Then we have proposed an interrogation span suitable to the analysis of an extraordinary vertex neighborhood. It is independent of the subdivision scheme used to synthesize the surface, thus we can compared several schemes. Then our work engaged in the polar spectral analysis of these geometric behaviours, taking into account there radial and angular characteristics regarding the topology of an extraordinary vertex neighbourhood. Our analysis extends the existing works for the interrogation of this modelisation tool. Secondly, we have proposed a description framework for the topological step of a subdivision scheme. Our system takes the form of a compact and flexible coding, it generalizes the existent descriptions. This coding allows the description of the topological step of all the known subdivision schemes, and many others.
Based on the properties of curvature connection and theory of differential geometry, a sufficient condition of curvature connection between two adjacent surfaces is obtained, and then a new method of curvature connection of surface patches around a common vertex is put forward by using these conditions and consistency. As a result, the obtained surface patches have the lowest degrees, whose degrees are five, and the relevant system of equations can be solved easily.
This paper aims at giving a self-contained description of the focal surfaces of the normal congruence of a triangular Bézier patch in terms of the control points of the patch. The normal congruence of a surface is an Euclidean concept and it is algebraic, if the original surface is algebraic. To calculate the parameter representation of the normal congruence of a triangular Bézier patch we need derivatives and normal vectors of the patch. For calculating the pair of so called focal points on each generator of the congruence, one has to investigate the curvature lines of the patch in addition. Thus the results become already of high algebraic order even for quadratic or cubic patches. Therefore the treatment is restricted to quadratic and cubic triangular Bézier patches and focal points of its normal congruence are calculated only for the normals at very special points of the patch: the corner points and the "midpoint". Therewith one can deduce control point systems for 'low' order approximations of the two patches of focal surfaces of the normal congruence. Finally the calculations are applied to a nu-merical example. The paper is a first attempt to deal with the focal surfaces of a line congruence explicitly and might deliver fundamentals to treat refraction congruences, which have applications in geometric optics.
The problem of trimming Bézier surfaces on Bézier surfaces contains many cases, such as the subdivision, conversion and conjoining.
Different methods have been given for some special cases. In this paper, by means of blossoming and parameter transformation,
a united approach is given for this problem. The approach can be extended to trim Bézier patches on any polynomial or rational
surfaces naturally.
L'objet de ce mémoire est l'analyse des schémas de subdivision, outil de modélisation de surface lisse multi-résolution. Nos travaux se sont tout d'abord consacrés à l'étude du comportement géométrique de ces surfaces au voisinage des sommets extraordinaires du maillage de contrôle. La géométrie d'une surface de subdivision présente un comportement complexe au voisinage de ces sommets, et ces effets parfois néfastes sont pour certains encore mal connus. Nous avons proposé un cadre d'évaluation du comportement géométrique d'une surface de subdivision à travers une mesure de qualité adaptée : le gradient de courbure absolue. Nous avons ensuite proposé un espace de visualisation adapté à l'analyse du voisinage d'un sommet extraordinaire.Celui-ci étant indépendant du schéma de subdivision utilisé, ce cadre d'analyse nous permet de les comparer. Nos travaux se sont alors portés sur une analyse fréquentielle polaire des comportements géométriques, en tenant compte de leurs caractéristiques radiales et angulaires par rapport à la topologie du voisinage d'un sommet extraordinaire. Notre analyse étend les études existantes pour l'évaluation des comportements géométriques de cet outil de modélisation. De plus, nous avons proposé un systàme de description de la phase de subdivision topologique d'un schéma de subdivision. Notre systàme prend la forme d'un codage compact et flexible, il généralise les descriptions existantes. Ce codage permet la description des phases topologiques de tous les schémas de subdivision connus, ainsi que de nombreuses autres.
An explicit formula for the conversion of a degree Bézier rectangular tensor product surface to a degree m + n Bézier triangle is derived. The control vertices of the Bézier triangle are shown to lie in the convex hull of the control vertices of the Bézier rectangle. Therefore the conversion formula is numerically stable. This formula is important because algorithms developed specifically for Bézier triangles can now also be used for surfaces composed of both Bézier rectangles and Bézier triangles.
There are two kinds of Bezier patches which are represented by different base functions, namely the triangular Bezier patch and the rectangular Bezier patch. In this paper, two results about these patches are obtained by employing functional compositions via shifting operators. One is the composition of a rectangular Bezier patch with a triangular Bezier function of degree #, the other one is the composition of a triangular Bezier patch with a rectangular Bezier function of degree # # #. The control points of the resultant patch in either case are the linear convex combinations of the control points of the original patch. With the shifting operators, the respective procedure becomes concise and intuitive. The potential applications about the two results include conversions between two kinds of Bezier patches, exact representation of a trimming surface, natural extension of original patches, etc. Key words rectangular Bezier patch, triangular Bezier patch, functional composition, com...
In view of the fundamental role that functional composition plays in mathematics, it is not surprising that a variety of problems in geometric modeling can be viewed as instances of the following composition problem: given representations for two functions F and G, compute a representation of the function H = F ffi G: We examine this problem in detail for the case when F and G are given in either B'ezier or B-spline form. Blossoming techniques are used to gain theoretical insight into the structure of the solution which is then used to develop efficient, tightly codable algorithms. From a practical point of view, if the composition algorithms are implemented as library routines, a number of geometric modeling problems can be solved with a small amount of additional software. This paper was published in TOG, April 1993, pg 113-135 Categories and Subject Descriptors: I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling - curve, surface, and object representations; J.6 [...
An efficient, robust algorithm is presented for converting a bivariate polynomial expressed in the power basis directly to its equivalent Bernstein (Bézier) form over an arbitrary triangular region. It represents an improvement over existing algorithms which, in the course of transforming the polynomial, may encounter degeneracy problems that must be handled with additional logic.
A simple algorithm is given which calculates the Bézier points of a surface over a quadrangle from the given generalized Bézier points of the surface over a triangle. An example is added.
This chapter outlines the history of the major developments in the area of curves and surfaces as they entered the area of computer aided geometric design (CAGD). The term “CAGD” was coined by R. Barnhill and R. Riesenfeld in 1974 when they organized a conference on that topic at the University of Utah. CAGD deals with the construction and representation of free-form curves, surfaces, or volumes. The earliest recorded use of curves in a manufacturing environment goes back to early AD Roman times for the purpose of shipbuilding. Shipbuilding connection was the earliest use of constructive geometry to define free-form shapes. Another early influential development for CAGD was the advent of numerical control (NC) in the 1950s. The chapter reveals that curves were employed by draftsmen for centuries; the majority of these curves were circles, but some were free-form. It defines French curves as wooden curves consisting of pieces of conies and spirals. The chapter also discusses the generalization of the concept of splines to surfaces.
The subdivision of Bézier triangles is studied from the viewpoint of multiaffine mappings. Using the principle that polynomials of degree n and symmetric n-affine maps are equivalent to each other, a theorem is derived that provides simultaneous closed form solutions for the various recursive algorithms due to Goldman [12]. From this, the major properties of subdivision algorithms for Bézier triangles are readily deduced.
In this article, the problems of visual continuity for rational Bézier surfaces are discussed. Concise conditions of first-order and second-order visual continuity for rational triangular Bézier surfaces and between rational triangular Bézier patches and rational rectangular Bézier patches are given. Meanwhile, the transformation formulae between rational triangular patches and rational rectangular patches are obtained.
We give an explicit formula for converting a triangular Bézier patch into three non-degenerate rectangular Bézier patches of the same degree. The use of certain operators simplifies the formulation of such a decomposition. The formula yields a stable recursive algorithm for computing the control points of the rectangular patches. We also illustrate the formula with an example.
In an earlier paper the author studied subdivision algorithms for Bézier curves. These algorithms were generated by constructing degenerate Bézier triangles and tetrahedra and using their edges to subdivide the original curve. In this paper we take up the study of subdivision algorithms for the triangular Bézier surfaces first introduced by Sabin. It will be shown that such algorithms can be generated by constructing higher dimensional degenerate Bézier simplices and using their triangular subsimplices to subdivide the original triangular surface.
Trimming of surfaces and volumes, curve and surface modeling via Bézier’s idea of destortion, segmentation, reparametrization, geometric continuity are examples of applications of functional composition. This paper shows how to compose polynomial and rational tensor product Bézier representations. The problem of composing Bézier splines and B-spline representations will also be addressed in this paper.
An explicit formula as well as a geometric algorithm are given for converting a rectangular subpatch of a triangular Bézier surface into a tensor product Bézier representation. Based on de Casteljau recursions and convex combinations of combinatorial constants, both formula and algorithm are numerically stable. Examples and special problems are discussed such as the suitcase corner problem and surfaces of the same polynomial degree.
This paper describes two algorithms for solving the following general problem: Given two polynomial maps f: Rn ↦ RN and S RN ↦ Rd in Bézier simplex form, find the composition map &Stilde; = S ° f in Bézier simplex form (typically, n ≤ N ≤ d ≤ 3). One algorithm is more appropriate for machine implementation, while the other provides somewhat more geometric intuition. The composition algorithms can be applied to the following problems: evaluation, subdivision, and polynomial reparameterization of Bézier simplexes; joining Bézier curves with geometric continuity of arbitrary order; and the determination of the control nets of Bézier curves and triangular Bézier surface patches after they have undergone free-form deformations.
Concerning subdivision of Bézier triangles
- W. Boehm
- G. Farin
- W. Boehm
- G. Farin
Subdivision of triangular Bézier patches into rectangular Bézier patches
- L. Lino
- J. Wilde
- L. Lino
- J. Wilde
Concerning subdivision of Bé triangles
- W Boehm
- G Farin
W. Boehm, G. Farin, Concerning subdivision of Bé triangles, Computer-Aided Design 15 (5) (1983) 260–261 (Letter to the Editor).
Subdivision of triangular Bé patches into rectangular Bé patches, Advances in Design Automation
- L Lino
- J Wilde
L. Lino, J. Wilde, Subdivision of triangular Bé patches into rectangular Bé patches, Advances in Design Automation 32 (2) (1991) 1–6.
Subdivision of triangular Bézier patches into rectangular Bézier patches Advances in design automation 32
- L Lino
- J Wilde
Concerning subdivision of Bézier triangles. Computer-Aided Design. v15 i5
- W Boehm
- G Farin
Subdivision of triangular Bézier patches into rectangular Bézier patches
- Lino
Concerning subdivision of Bézier triangles
- Boehm