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Fundamentals of computer aided geometric design. Translated from the German by Larry L. Schumaker
... Let us assume that we have two objects r 1 and r 2 where r 1 can be a NURBS surface patch [11,20] or scanned data points in a different pose and no scaling effects are involved, and r 2 is a NURBS surface patch. The Euclidean distance between two points p 1 and p 2 is defined as ...
... This can be achieved by various methods such as the circle fitting method by Martin [17], the CP method by Stokely and Wu [24] and the CT method by Sander and Zucker [21]. At m i ; we assume we have computed estimates for the K i and H i ; where i ¼ 1; 2; 3: Next, we subdivide r 2 into rational Bézier surface patches B j ðj ¼ 1; …; nÞ by inserting appropriate knots [11,20]. Then for each rational Bézier surface patch B j ; we express K j and H j in the bivariate rational Bernstein polynomial basis using Eq. ...
A novel method of matching for 3-D free-form objects (points vs. surface and surface vs. surface) is proposed. The method formulates the problem in terms of solution of a non-linear polynomial equation system, which can be solved robustly by the Interval Projected Polyhedron (IPP) algorithm. Two intrinsic surface properties, the Gaussian and the mean curvatures, are used as object features for matching. The related iso-curvature lines are used to establish the correspondence between two objects. The intersection points of these iso-curvature lines are calculated and sorted out to satisfy the Euclidean constraints from which the translation and rotation transformations are estimated. The performance of the proposed algorithm is also analyzed. This approach can cover global and partial matching, and be applied to automated inspection, copyright protection of NURBS models, and object recognition. Examples illustrate our technique.
... The algorithm can be extended for C 0-continuous curves and possibly for open curves. Another potential application for using the theories that have been developed in this paper could be on blending of curves (see Chapter 14 in [29]). The application areas are being explored at present. ...
Of late, researchers appear to be intrigued with the question; Given a set of points, what is the region occupied by them? The answer appears to be neither straight forward nor unique. Convex hull, which gives a convex enclosure of the given set, concave hull, which generates non-convex polygons and other variants such as αα-hull, poly hull, rr-shape and ss-shape etc. have been proposed. In this paper, we extend the question of finding a minimum area enclosure (MAE) to a set of closed planar freeform curves, not resorting to sampling them. An algorithm to compute MAE has also been presented. The curves are represented as NURBS (non-uniform rational B-splines). We also extend the notion of αα-hull of a point set to the set of closed curves and explore the relation between alpha hull (using negative alpha) and the MAE.
... Parametric cubic curve segments are widely used in computer-aided design (CAD) and computeraided geometric design (CAGD) applications. They commonly occur as BÃ ezier or B-spline representations [4,6]. They are popular mainly because ...
Parametric polynomial cubic curve segments are widely used in computer-aided design and computer-aided geometric design applications because their flexibility makes them suitable for use in the interactive design of curves and surfaces. Shape properties of these segments have been studied and results on the occurrence of cusps, and whether the segment is S- or C-shaped, are available in the literature. Their critical points, however, are not as well known. This paper presents results on the number and location of curvature extrema of these segments. All possible numbers of curvature extrema are found and it is demonstrated that all possible curvature extrema can be obtained numerically for any planar parametric cubic curve. These results are useful in the study of the fairness of curves designed with parametric polynomial cubic curve segments.
... We will adopt the notation from [16]. For a precise and detailed theoretical background on B-splines and NURBS in computer aided geometric design we refer the reader to [28][29][30]. ...
Isogeometric analysis is a numerical simulation method which uses the NURBS based representation of CAD models. NURBS stands for non-uniform rational B-splines and is a generalization of the concept of B-splines. The isogeometric method uses the tensor product structure of 2-or 3-dimensional NURBS functions to parameterize domains, which are structurally equivalent to a rectangle or a hexahedron. The special case of singularly parameterized NURBS surfaces and NURBS volumes is used to represent non-quadrangular or non-hexahedral domains without splitting, which leads to a very compact and convenient representation. If the parameterization of the physical domain is available, the test functions for the Isogeometric Analysis are obtained by composing the inverse of the domain parameterization with the NURBS basis functions. In the case of singular parameterizations, however, some of the resulting test functions are not well defined at the singular points and they do not necessarily satisfy the required integrability assumptions. Consequently, the stiffness matrix integrals which occur in the numerical discretizations may not exist. After summarizing the basics of the isogeometric method, we discuss the existence of the stiffness matrix integrals for 1-, 2- and 3-dimensional second order elliptic partial differential equations. We consider several types of singularities of NURBS parameterizations and derive conditions which guarantee the existence of the required integrals. In addition, we present cases with diverging integrals and show how to modify the test functions in these situations.
... While freeform surfaces have flexibility and allow the manipulations typically required in the conceptual design phase, they lacked the ability to express design intent or knowledge in a detailed and explicit manner. Although low-level shape parameters such as weights, knots, and control points [16] are available to adjust the freeform surfaces, they are counter-intuitive to designers. Many shape deformation methods have been developed in the last decade to increase the intuitiveness of the freeform shape deformation, and to increase the designers' ability to control the shape changes through mesh and surface deformation. ...
Today’s product designer is being asked to develop high quality, innovative products at an ever increasing pace. To meet this need, an intensive search is underway for advanced design methodologies that facilitate the acquisition of design knowledge and creative ideas for later reuse. Additionally, designers are embracing a wide range of 3D digital design applications, such as 3D digitization, 3D CAD and CAID, reverse engineering (RE), CAE analysis and rapid prototyping (RP). In this paper, we propose a reverse engineering innovative design methodology called Reverse Innovative Design (RID). The RID methodology facilitates design and knowledge reuse by leveraging 3D digital design applications. The core of our RID methodology is the definition and construction of feature-based parametric solid models from scanned data. The solid model is constructed with feature data to allow for design modification and iteration. Such a construction is well suited for downstream analysis and rapid prototyping. In this paper, we will review the commercial availability and technological developments of some relevant 3D digital design applications. We will then introduce three RE modelling strategies: an autosurfacing strategy for organic shapes; a solid modelling strategy with feature recognition and surface fitting for analytical models; and a curve-based modelling strategy for accurate reverse modelling. Freeform shapes are appearing with more frequency in product development. Since their “natural” parameters are hard to define and extract, we propose construction of a feature skeleton based upon industrial or regional standards or by user interaction. Global and local product definition parameters are then linked to the feature skeleton. Design modification is performed by solving a constrained optimization problem. A RID platform has been developed and the main RE strategies and core algorithms have been integrated into SolidWorks as an add-in product called ScanTo3D. We will use this system to demonstrate our RID methodology on a collection of innovative consumer product design examples.
In the Isogeometric Analysis (IGA) framework, the computational domain has very often a multipatch representation. The multipatch domain can be obtained by a volume segmentation of a boundary represented domain, e.g., provided by a Computer Aided Design (CAD) model. Typically, small gap and overlapping regions can appear at the patch interfaces of such multipatch representations. In the current work we consider multipatch representations having only small overlapping regions between the patches. We develop a Discontinuous Galerkin (DG)- IGA method which can be immediately applied to these representations. Our method appropriately connects the fluxes of the one face of the overlapping region with the flux of the opposite face. We provide a theoretical justification of our approach by splitting the whole error into two components: the first is related to the incorrect representation of the patches (consistency error) and the second to the approximation properties of the IGA space. We show bounds for both components of the error. We verify the theoretical error estimates in a series of numerical examples.
High-order methods are increasingly popular in computational fluid dynamics, but the construction of suitable curvilinear meshes still remains a challenge. This paper presents a strictly local optimization method to construct high-order triangular surface patches of high quality and accuracy. It combines fitting and energy-minimization, in which approximate bending and stretching functionals are minimized by means of an incremental procedure. The method was applied to analytically defined smooth surfaces as well as scattered surface data derived from scanning data. In both cases the optimization yielded considerable improvements in patch quality, while preserving the accuracy of pure least-squares fitting. As intended, the method achieves the greatest benefit with coarse meshes and high polynomial order.
This paper develops an algorithm for constructing C
3 continuous rational quartic B-spline curves such that each rational Bézier curve segment corresponds to a two-harmonic trajectory pattern. Such smooth low-harmonic spline curves can be used for synthesizing robot joint trajectories that are least susceptible to vibrational excitation.
The Silences of the Archives, the Reknown of the Story.
The Martin Guerre affair has been told many times since Jean de Coras and Guillaume Lesueur published their stories in 1561. It is in many ways a perfect intrigue with uncanny resemblance, persuasive deception and a surprizing end when the two Martin stood face to face, memory to memory, before captivated judges and a guilty feeling Bertrande de Rols. The historian wanted to go beyond the known story in order to discover the world of the heroes. This research led to disappointments and surprizes as documents were discovered concerning the environment of Artigat’s inhabitants and bearing directly on the main characters thanks to notarial contracts. Along the way, study of the works of Coras and Lesueur took a new direction. Coming back to the affair a quarter century later did not result in finding new documents (some are perhaps still buried in Spanish archives), but by going back over her tracks, the historian could only be struck by the silences of the archives that refuse to reveal their secrets and, at the same time, by the possible openings they suggest, by the intuition that almost invisible threads link here and there characters and events.
We prove that, in contrast to the case for rational surfaces, some tensor product representations through spaces containing algebraic, trigonometric and hyperbolic polynomials are monotonicity preserving. The surface representations provided in this paper are the only known monotonicity preserving surfaces in addition to the tensor product Bézier and tensor product B-spline surfaces.
We propose a novel approach to the problem of multi-degree reduction of Bézier triangular patches with prescribed boundary control points. We observe that the solution can be given in terms of bivariate dual discrete Bernstein polynomials. The algorithm is very efficient thanks to using the recursive properties of these polynomials. The complexity of the method is O(n2m2)O(n2m2), nn and mm being the degrees of the input and output Bézier surfaces, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined triangular Bézier surfaces, the result is a composite surface of global CrCr continuity with a prescribed order rr. Some illustrative examples are given.
One hundred years after the introduction of the Bernstein polynomial basis, we survey the historical development and current state of theory, algorithms, and applications associated with this remarkable method of representing polynomials over finite domains. Originally introduced by Sergei Natanovich Bernstein to facilitate a constructive proof of the Weierstrass approximation theorem, the leisurely convergence rate of Bernstein polynomial approximations to continuous functions caused them to languish in obscurity, pending the advent of digital computers. With the desire to exploit the power of computers for geometric design applications, however, the Bernstein form began to enjoy widespread use as a versatile means of intuitively constructing and manipulating geometric shapes, spurring further development of basic theory, simple and efficient recursive algorithms, recognition of its excellent numerical stability properties, and an increasing diversification of its repertoire of applications. This survey provides a brief historical perspective on the evolution of the Bernstein polynomial basis, and a synopsis of the current state of associated algorithms and applications.
We propose an efficient approach to the problem of multi-degree reduction of rectangular Bézier patches, with prescribed boundary control points. We observe that the solution can be given in terms of constrained bivariate dual Bernstein polynomials. The complexity of the method is O(mn1n2)O(mn1n2) with m ≔ min(m1, m2), where (n1, n2) and (m1, m2) is the degree of the input and output Bézier surface, respectively. If the approximation—with appropriate boundary constraints—is performed for each patch of several smoothly joined rectangular Bézier surfaces, the result is a composite surface of global Cr continuity with a prescribed r ⩾ 0. In the detailed discussion, we restrict ourselves to r ∈ {0, 1}, which is the most important case in practical application. Some illustrative examples are given.
This paper presents the use of Powell–Sabin splines in the context of isogeometric analysis for the numerical solution of advection–diffusion–reaction equations. Powell–Sabin splines are piecewise quadratic C1 functions defined on a given triangulation with a particular macro-structure. We discuss the Galerkin discretization based on a normalized Powell–Sabin B-spline basis. We focus on the accurate detection of internal and boundary layers, and on local refinements. We apply the approach to several test problems, and we illustrate its effectiveness by a comparison with classical finite element and recent isogeometric analysis procedures.
. Univariate cubic L
1 splines provide C
1-smooth, shape-preserving interpolation of arbitrary data, including data with abrupt changes in spacing and magnitude. The
minimization principle for univariate cubic L
1 splines results in a nondifferentiable convex optimization problem. In order to provide theoretical treatment and to develop
efficient algorithms, this problem is reformulated as a generalized geometric programming problem. A geometric dual with a
linear objective function and convex quadratic constraints is derived. A linear system for dual to primal conversion is established.
The results of computational experiments are presented. In the natural norm for this class of problems, namely, the L
1 norm of the second derivative, the geometric programming approach finds better solutions than the previously used discretization
method.
We present algorithms for computing the differential geometry properties of intersection curves of three implicit surfaces in R4, using the implicit function theorem and generalizing the method of X. Ye and T. Maekawa for 4-dimension. We derive t,n,b1,b2 vectors and curvatures (k1,k2,k3) for transversal intersections of the intersection problem.
In parametric curve interpolation there is given a sequence of data points and corresponding parameter values (nodes), and we want to find a parametric curve that passes through data points at the associated parameter values. We consider those interpolating curves that are described by the combination of control points and blending functions. We study paths of control points and points of the interpolating curve obtained by the alteration of one node. We show geometric properties of quadratic Bézier interpolating curves with uniform and centripetal parameterizations. Finally, we propose geometric methods for the interactive modification and specification of nodes for interpolating Bézier curves.
In this paper, we derive the bounds on the magnitude of lth (l=2,3) order derivatives of rational Bézier curves, estimate the error, in the L∞ norm sense, for the hybrid polynomial approximation of the lth (l=1,2,3) order derivatives of rational Bézier curves. We then prove that when the hybrid polynomial approximation converges to a given rational Bézier curve, the lth (l=1,2,3) derivatives of the hybrid polynomial approximation curve also uniformly converge to the corresponding derivatives of the rational curve. These results are useful for designing simpler algorithms for computing tangent vector, curvature vector and torsion vector of rational Bézier curves.
In this article, we address the problem of interpolating data points by regular L1-spline polynomial curves of smoothness Ck, k⩾1, that are invariant under rotation of the data. To obtain a C1 cubic interpolating curve, we use a local minimization method in parallel on five data points belonging to a sliding window. This procedure is extended to create Ck-continuous L1 splines, k⩾2, on larger windows. We show that, in the Ck-continuous (k⩾1) interpolation case, this local minimization method preserves the linear parts of the data well, while a global L1 minimization method does not in general do so. The computational complexity of the procedure is linear in the global number of data points, no matter what the order Ck of smoothness of the curve is.Research highlights► Shape preserving interpolation of data points by L1 polynomial spline curves. ► Five-point window interpolation. ► Exact calculation of the nonlinear minimization problem.
This paper describes a method for joining two circles with an S-shaped or with a broken back C-shaped transition curve, composed of at most two spiral segments. In highway and railway route design or car-like robot path planning, it is often desirable to have such a transition. It is shown that a single cubic curve can be used for blending or for a transition curve preserving G2 continuity with local shape control parameter and more flexible constraints. Provision of the shape parameter and flexibility provide freedom to modify the shape in a stable manner which is an advantage over previous work by Meek, Walton, Sakai and Habib.
In this paper, the issue of multi-degree reduction of Bézier curves with C1 and G2-continuity at the end points of the curve is considered. An iterative method, which is the first of this type, is derived. It is shown that this algorithm converges and can be applied iteratively to get the required accuracy. Some examples and figures are given to demonstrate the efficiency of this method.
Piecewise cubic and quartic polynomial curves with adjustable interpolation points are presented in this paper. The adjustable interpolation points are represented by local shape parameters and the given control points. Based on the choice of endpoint tangents of curve segments, piecewise cubic C1, piecewise cubic G2 and piecewise quartic C2 curves are given. The representations of the piecewise cubic C1 curves and the piecewise quartic C2 curves are integrated representations of approximating and interpolating curves. By changing the values of the local shape parameters, local approximating curves and local interpolating curves can be generated respectively.
Backward stability of the Casteljau algorithm and two more efficient algorithms for polynomial tensor product surfaces with interest in CAGD is shown. The conditioning of the corresponding bases are compared. These algorithms are also compared with the corresponding Horner algorithm and their higher accuracy is shown. A running error analysis of the algorithms is also carried out providing algorithms which calculate “a posteriori” sharp error bounds simultaneously to the evaluation of the surface without increasing significantly the computational cost.
Isogeometric analysis is a new method for the numerical simulation of problems governed by partial differential equations. It possesses many features in common with finite element methods (FEM) but takes some inspiration from Computer Aided Design tools. We illustrate how quasi-interpolation methods can be suitably used to set Dirichlet boundary conditions in isogeometric analysis. In particular, we focus on quasi-interpolant projectors for generalized B-splines, which have been recently proposed as a possible alternative to NURBS in isogeometric analysis.
Virtual prototyping (VP) typically involves analyzing computer aided design (CAD) models for different end applications such as manufacturability or assemblability analysis. Such analyses may be performed via the aid of expert system agents which could be residing in a distributed fashion on the Internet. Each expert system agent/module requires CAD information at a certain level of geometric abstraction and outputs the analysis at a different level of abstraction. Therefore, a basis is needed to define these different levels of geometric abstraction. The current paper defines and analyzes these different levels of geometric abstraction in the context of an architecture for achieving VP via a central broker that interacts with these different modules—each providing a different level of abstraction.
In CAGD curves are described mostly by means of the combination of control points and basis functions. If we associate weights with basis functions and normalize them by their weighted sum, we obtain another set of basis functions that we call quotient bases. We show some common characteristics of curves defined by such quotient basis functions. Following this approach we specify the rational counterpart of the recently introduced cyclic basis, and provide a ready to use tool for control point based exact description of a class of closed rational trigonometric curves and surfaces. We also present the exact control point based description of some famous curves (Lemniscate of Bernoulli, Zhukovsky airfoil profile) and surfaces (Dupin cyclide and the smooth transition between the Boy surface and the Roman surface of Steiner) to illustrate the usefulness of the proposed tool.
We construct biorthogonal spline wavelets for periodic splines which extend the notion of “lazy” wavelets for linear functions (where the wavelets are simply a subset of the scaling functions) to splines of higher degree. We then use the lifting scheme in order to improve the approximation properties with respect to a norm induced by a weighted inner product with a piecewise constant weight function. Using the lifted wavelets we define a multiresolution analysis of tensor-product spline functions and apply it to image compression of black-and-white images. By performing–as a model problem–image compression with black-and-white images, we demonstrate that the use of a weight function allows to adapt the norm to the specific problem.
Many problems in computer aided geometric design and geometry processing are stated as least-squares optimizations. Least-squares problems are well studied and widely used but exhibit immanent drawbacks such as high sensitivity to outliers. For this reason, we consider techniques for the registration of point clouds and surface fitting to point sets based on the l1-norm. We develop algorithms to solve l1-registration and l1-fitting problems and explore the emerging non-smooth minimization problems. We describe efficient ways to solve the optimization programs and present results for various applications.
In this paper we present a very efficient Hermite subdivision scheme, based on rational functions, and outline its potential applications, with special emphasis on the construction of cubic-like B-splines — well suited for the design of constrained curves and surfaces.
In the present paper we investigate rational two-parameter families of spheres and their envelope surfaces in Euclidean R3. The four dimensional cyclographic model of the set of spheres in R3 is an appropriate framework to show that a quadratic triangular Bézier patch in R4 corresponds to a two-parameter family of spheres with rational envelope surface. The construction shows also that the envelope has rational offsets. Further we outline how to generalize the construction to obtain a much larger class of surfaces with similar properties.
In [Alcazar, J.G., Sendra, J.R. 2006. Local shape of offsets to rational algebraic curves. Tech. Report SFB 2006-22 (RICAM, Austria); Alcazar, J.G., Sendra, J.R. 2007. Local shape of offsets to algebraic curves. Journal of Symbolic Computation 42, 338–351], the notion of good local behavior of an offset to an algebraic curve was introduced to mean that the topological behavior of the offset curve was locally good, i.e. that the shape of the starting curve and of its offset were locally the same. Here, we introduce the notion of good global behavior to describe that the offset behaves globally well, from a topological point of view, so that it can be decomposed as the union of two curves (maybe not algebraic) each one with the topology of the starting curve. We relate this notion with that of good local behavior, and we give sufficient conditions for the existence of an interval of distances (0,γ) such that for all d∈(0,γ) the topological behavior of the offset Od(C) is both locally and globally nice. A similar analysis for the trimmed offset is also done.
This paper proposes a second order geometric iteration algorithm for point projection and inversion on parametric surfaces. The iteration starts from an initial projection estimation. In each iteration, we construct a second order osculating torus patch to the parametric surface at the previous projection. Then we project the test point onto the torus patch to compute the next projection and its parameter. This iterative process is terminated when the parameter satisfies the required precision. Experiments demonstrate the convergence speed of our algorithm.
In this paper we present a new class of spline functions with tension properties. These splines are composed by polynomial cubic pieces and therefore are conformal to the standard, NURBS based CAD/CAM systems.
A new formulation for the representation and designing of curves and surfaces is presented. It is a novel generalization of Bézier curves and surfaces. Firstly, a class of polynomial basis functions with n adjustable shape parameters is present. It is a natural extension to classical Bernstein basis functions. The corresponding Bézier curves and surfaces, the so-called Quasi-Bézier (i.e., Q-Bézier, for short) curves and surfaces, are also constructed and their properties studied. It has been shown that the main advantage compared to the ordinary Bézier curves and surfaces is that after inputting a set of control points and values of newly introduced n shape parameters, the desired curve or surface can be flexibly chosen from a set of curves or surfaces which differ either locally or globally by suitably modifying the values of the shape parameters, when the control polygon is maintained. The Q-Bézier curve and surface inherit the most properties of Bézier curve and surface and can be more approximated to the control polygon. It is visible that the properties of end-points on Q-Bézier curve and surface can be locally controlled by these shape parameters. Some examples are given by figures.
The aim of this paper is to present a new class of B-spline-like functions with tension properties. The main feature of these basis functions consists in possessing C3 or even C4 continuity and, at the same time, being endowed by shape parameters that can be easily handled. Therefore they constitute a useful tool for the construction of curves satisfying some prescribed shape constraints. The construction is based on a geometric approach which uses parametric curves with piecewise quintic components.
Approximating complex curves with simple parametric curves is widely used in
CAGD, CG, and CNC. This paper presents an algorithm to compute a certified
approximation to a given parametric space curve with cubic B-spline curves. By
certified, we mean that the approximation can approximate the given curve to
any given precision and preserve the geometric features of the given curve such
as the topology, singular points, etc. The approximated curve is divided into
segments called quasi-cubic B\'{e}zier curve segments which have properties
similar to a cubic rational B\'{e}zier curve. And the approximate curve is
naturally constructed as the associated cubic rational B\'{e}zier curve of the
control tetrahedron of a quasi-cubic curve. A novel optimization method is
proposed to select proper weights in the cubic rational B\'{e}zier curve to
approximate the given curve. The error of the approximation is controlled by
the size of its tetrahedron, which converges to zero by subdividing the curve
segments. As an application, approximate implicit equations of the approximated
curves can be computed. Experiments show that the method can approximate space
curves of high degrees with high precision and very few cubic B\'{e}zier curve
segments.
We present an approach of computing the intersection curve of
two rational parametric surface and , one being
projectable and hence can easily be implicitized. Plugging the parametric
surface to the implicit surface yields a plane algebraic curve G(v,t)=0. By
analyzing the topology graph \G of G(v,t)=0 and the singular points on the
intersection curve we associate a space topology graph to
, which is homeomorphic to and therefore leads us to
an approximation for in a given precision.
By introducing the inner-product matrix of two vector functions and using conversion matrix, explicit formulas for the dual basis functions of Wang–Bézier type generalized Ball bases (WBGB) with respect to the Jacobi weight function are given. The dual basis functions with and without boundary constraints are also considered. As a result, the paper includes the weighted dual basis functions of Bernstein basis, Wang–Ball basis and some intermediate bases. Dual functionals for WBGB and the least square approximation polynomials are also obtained.
In the present paper we investigate the approximation of the localized Bernstein polynomials to continuous functions by using the central limit theorem of probability theory. Some new pointwise approximation theorems are obtained.
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.
Error analysis of the usual method to evaluate rational Bézier surfaces is performed. The corresponding running error analysis is also carried out and the sharpness of our running error bounds is shown. We also modify the evaluation algorithm to include such error bounds without increasing significantly its computational cost.
This paper addresses the problem of shape preserving spline interpolation formulated as a differential multipoint boundary value problem (DMBVP for short). Its discretization by mesh method yields a five-diagonal linear system which can be ill-conditioned for unequally spaced data. Using the superposition principle we split this system in a set of tridiagonal linear systems with a diagonal dominance. The latter ones can be stably solved either by direct (Gaussian elimination) or iterative methods (SOR method and finite-difference schemes in fractional steps) and admit effective parallelization. Numerical examples illustrate the main features of this approach.
Running error analysis of the corner cutting algorithm for rational Bézier surfaces is carried out and the sharpness of the corresponding error bounds is shown.
We propose a parametric tensioned version of the FVS macro-element to control the shape of the composite surface and remove artificial oscillations, bumps and other undesired behaviour. In particular, this approach is applied to C1 cubic spline surfaces over a four-directional mesh produced by two-stage scattered data fitting methods.
We propose a new combination of the bivariate Shepard operators (Coman and Trîmbiţaş, 2001 [2]) by the three point Lidstone polynomials introduced in Costabile and Dell’Accio (2005) [7]. The new combination inherits both degree of exactness and Lidstone interpolation conditions at each node, which characterize the interpolation polynomial. These new operators find application to the scattered data interpolation problem when supplementary second order derivative data are given (Kraaijpoel and van Leeuwen, 2010 [13]). Numerical comparison with other well known combinations is presented.
In the paper [A. Rababah, M. Alqudah, Jacobi-weighted orthogonal polynomials on triangular domains, J. Appl. Math. 3 (2005) 205-217.], Jacobi-weighted orthogonal polynomials P"n","r^(^@a^,^@b^,^@c^)(u,v,w),@a,@b,@c>-1 on the triangular domain T for values of @a,@b,@c>-1 in the plane @a+@b+@c=0 are constructed. In this paper, the results are generalized to every point in the space @?@a,@b,@c>-1.
A cubic trigonometric Bézier curve analogous to the cubic Bézier curve, with two shape parameters, is presented in this work. The shape of the curve can be adjusted by altering the values of shape parameters while the control polygon is kept unchanged. With the shape parameters, the cubic trigonometric Bézier curves can be made close to the cubic Bézier curves or closer to the given control polygon than the cubic Bézier curves. The ellipses can be represented exactly using cubic trigonometric Bézier curves.
In this paper we present a local fitting algorithm for converting smooth planar curves to B-splines. For a smooth planar curve a set of points together with their tangent vectors are first sampled from the curve such that the connected polygon approximates the curve with high accuracy and inflexions are detected by the sampled data efficiently. Then, a G1 continuous Bézier spline curve is obtained by fitting the sampled data with shape preservation as well as within a prescribed accuracy. Finally, the Bézier spline is merged into a C2 continuous B-spline curve by subdivision and control points adjustment. The merging is guaranteed to be within another error bound and with no more inflexions than the Bézier spline. In addition to shape preserving and error control, this conversion algorithm also benefits that the knots are selected automatically and adaptively according to local shape and error bound. A few experimental results are included to demonstrate the validity and efficiency of the algorithm.
We present the equivolumetric offset surface which is a special kind of variable distance offset surfaces. For a given base surface, the volume bounded by the base surface and its equidistance offset may vary depending on the curvature of the base surface. The offset distance function of the equivolumetric offset surface is carefully chosen to compensate this curvature effect and to equalize the volumetric ratio. The explicit formulation of the equivolumetric offset distance function is given by using the Chebyshev cube root.
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