No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Rationale Bézier—Volumina
Abstract
Die vom CAGD bekannten Techniken für rationale Freiform Bézier- Kurven und -Flächen werden auf rationale trivariate Darstellungen erweitert. Rationale Bézier-Volumina gestatten die Beschreibung von Freiform-Volumina als auch die exakte Darstellung von 3D-Primitiven, wie Kugel, Kegel, Torus, etc. Techniken zur Visualisierung rationaler Bézier-Volumina werden angesprochen.
An algorithm that creates planar and arbitrarily curved sections of free-form volumes is presented. The definition of free-form volumes generalizes techniques from free-form curves and surfaces to trivariate representation. The definition is given for volumes in the Bernstein-Bezier representation. The author illustrates an intersection algorithm that can be used to perform intersection operations on free-form volumes. Some calculated examples are given. The algorithm can be used as a subroutine for algorithms which are able to perform more general intersections of free-form volumes, e.g. Boolean operations on two free-form volumes
The two most popular methods of representing solids are the Constructive Solid Geometry CSG and the Boundary Representation BR (see e.g. [Casale 85]). CSG represents a solid as a combination of solid primitives such as blocks, cones, and spheres through Boolean operators. BR represents a solid by the description of the bounding surface elements of the solid, the edges bordering adjacent surface elements and the vertices where such edges meet. Each of the two methods has advantages and shortcomings. Two disadvantages might be that the design, the free-form character of both methods is not very rich and that they assume internal homogeneity. But sometimes there is the need for a free-form modeling method and sometimes we are interested not only in the surface of a solid but also in his interior structure, and even if interior information is not desired the solid definition can serve as a useful tool in many geometric operations. Therefore there is a wide variety of volume, i.e. trivariate representation applications, e.g. the temperature, the pressure, the gravitational or the electromagnatic field, etc. as a function of the three spatial variables, the motion of a (changing) surface, e.g. the diffusion of the surface of a chemical reaction, the description of inhomogeneous material, surface generation by geometric operations or as contours of trivariate hypersurfaces [Sederberg 85, 86] and modification [Casale 85], [Farouki 85], [Sederberg 86], etc. (for details and more examples see the literature listed in the references).
We construct four-dimensional surfaces that approximate arbitrarily placed information. The basis strategy is (1) interpolate to the arbitrarily placed data with a general method, (2) then evaluate this general method over a coarse rectilinear grid to provide data for a tensor product method, and (3) render the surface by evaluating the tensor product over a finte grid. Graphical illustrations are also included.
This article is both a survey and a tutorial on the theory and use of rational Bézier curves and rational B-spline curves, as well as the corresponding surface schemes.
‘Surfaces in Computer Aided Geometric Design’ focuses on the representation and design of surfaces in a computer graphics environment. This new area has the dual attractions of interesting research problems and important applications. The subject can be approached from two points of view: The design of surfaces which includes the interactive modification of geometric information and the representation of surfaces for which the geometric information is relatively fixed. Design takes place in 3-space whereas representation can be higher dimensional. ‘Surfaces in CAGD’ can be traced from its inception in rectangular Coons patches and Bezier patches to triangular patches which are current research topics. Triangular patches can interpolate and approximate to arbitrarily located data and require the preprocessing steps of triangulation and derivative estimation. New contouring methods have been found using these triangular patches. Finally, multidimensional interpolation schemes have been based on tetrahedral interpolants and are illustrated by surfaces in 4-space by means of color computer graphics.
We construct a symmetric rational quartic map from the standard triangle onto an octant of a sphere. The surface is non-degenerate: all Bézier points are distinct and their associated weights are positive.
This paper looks at the problem of efficiently representing curves and surfaces used frequently in engineering design. An approach is presented whereby the capability of the rational Bézier- and B-spline schemes of being able to apply infinite control points is exploited. It is shown that algorithms used to compute rational curves and surfaces are not sensitive to the type of control points, i.e., finite or infinite. It is also pointed out that the presence of infinity simplifies the data set and results in a very flexible design scheme capable of representing complicated shapes in a relatively simple manner. Although the application of infinite control points involves the use of projective geometry, a 3-D interpretation is also presented to provide a design tool requiring no mathematical expertise in projective methods
A recursive algorithm for the evaluation of rational Bézier curves is presented; it consists of a construction that works with a constant cross ratio. This geometric principle is carried over to other algorithms.
A technique is presented for modelling with piecewise algebraic surface patches. Each surface patch is defined within a tetrahedral lattice of control points. The shape of the surface is modified by adjusting the weights of the control points. This scheme makes it possible to piece together two algebraic surface patches with any degree of cross-boundary derivative continuity. The application of piecewise algebraic surface patches to solid modelling is discussed.
Given a set of 3-D or 4-D scattered data, methods are presented that yield a bivariate or trivariate function that interpolate or approximate the given data. The subroutine package includes Hardy's multiquadric interpolant and multistage methods with many options available to the user, thus several different interpolation and approximation methods can be generated. The options available in the multistage methods can be effectively used if the data set is noisy, rapidly varying or nonuniformly distributed. The computational and storage requirements for the multistage methods are linear in the number of data points, thus they are efficient on large data sets.
A technique is presented for the graphical representation of some contour (level) surfaces of a function of three variables defined by its values on an array of points (x//i, y//j, z//k). The algorithm involves drawing and projecting the contour curves corresponding to the cross sections x equals x//i and y equals Y//j. Because the curves are drawn as thick, opaque bands, which mask all bands in the immediate background, a sense of depth is retained, yet background contours are still visible between the bands.
An adaptive method of contouring a bivariate surface defined over arbitrarily spaced data in two dimensions is introduced. The method has two stages: 1. Produce a Bézier polynomial surface defined over a union of triangles. 2. Use this surface to compute the required contour levels. Using positional data only, this method produces a C1-continuous surface and contours which can be made arbitrarily close to being C1-continuous. The contour is represented as a piecewise linear approximation to the C1-continuous contour.The method computes a linear representation of the surface in the neighborhood of the contour using an adaptive degree reduction algorithm. This algorithm is used to minimize the number of subdivisions necessary to obtain the desired smoothness of the contour. As a result of this adaptive subdivision, no contours are missed due to the coarseness of the original data points.
Two problems of the tensor product description of spatial domains by the method of Bernstein-Bézier are discussed: The construction of a volume point together with the first derivatives in this point and the solution of the interpolation problem. Some illustrative examples are given.
Elsevier Science Publishers B. V. (North Holland), 1986. vol. 7: includes a short bibliography. The rational Bezier curve and surface scheme of computer aided design is investigated from a geometric point of view. The investigation provides an insight into the inherent properties of the scheme making the use of rational functions for curve/surface representation and design easier
System requirements: IBM and compatibles with DOS 2.0 or higher or UNIX. This book offers an introduction to the field that emphasizes Bernstein-Bezier methods and presents subjects in an informal, readable style, making this an ideal text for an introductory course at the advanced undergraduate or graduate level. This 3rd edition includes several new section and numerical examples, a treatment of the new blossoming principle, and new C programs. All C programs are available on a disk included with the book. The Problems Sections at the end of each chapter have also been extended.
Diverse and powerful geometric representation schemes arising from the parametric polynomial formalism are surveyed. The authors emphasize throughout a unified perspective, indicating the status of individual formulations in a generalized scheme and illustrating the use of mutual transformations between various representations. The problems of geometric modeling are seen as falling into two broad categories: those of geometric representation, and those of geometric computation. The goal of most geometric computations is, in fact, the derivation of explicit geometric representations from complete but implicit geometric input data. The geometric representation schemes that have arisen in recent years, as briefly and incompletely summarized here, are seen as providing a workable solution to the first category of geometric problems. It is the second category, that of geometric computation, that now requires full attention. The issues to be addressed are inherently complex; involving high-order surface intersections, trimmed surface representations, geometric and topological consistency measures, and intelligent geometric interrogation procedures. The analytical and numerical tools used must therefore be correspondingly more sophisticated. 68 refs.
Analytic Solid Modeling (ASM) was develped as an aid for the engineer in the arduous task of modeling the complex geometry commonly found in mechanical engineering designs and has grown into commercial programs. This article describes the basics of ASM methodology, discusses its range of applications, and mentions some of the ongoing efforts that promise to extend the method into new application areas in the near future. Other popular and well-known solid-modeling forms are used such as constructive solid geometry and boundary representation, as frames of reference. Exotic applications are considered with this geometric modeling system because of the generality of its mathematical framework. The method extended to include the modeling of nongeometric data.
Darstellung 2-und 3-dimen-sionaler Strömungen. Rheinische Friedrich-Wilhelms
- H Duvenbeck
- A Schmidt
Adaptive Contouring of a trivariate interpolant Geometric Modeling, Algorithms and New Trends
- C S Petersen
- B R Piper
- A J Worsey
The Finite Element Method
- O C Zienkiewicz
- O Zienkiewicz
Eine neuer Aspekt des de Casteljau Algorithmus
- D Lasser
Finite Element Methods in CAD: Electrical and Magnetic Fields
- J C Sabonnadiere
- J L Coulomb
Definition of Solid Primitives by Rational Bezier Volumes
- D Lasser
- P Kirchgeßner
Darstellung 2- und 3-dimen- sionaler Strömungen
- H Duvenbeck
- A Schmidt
Der barycentrische Calcul. (1827) in Möbius, A. F.: Gesammelte Werke, Band I
- A F Möbius
- AF Möbius