No full-text available
To read the full-text of this research,
you can request a copy directly from the author.
Abstract
Free form volumes in rational Bézier representation are derived via homogeneous coordinates. Some properties and constructions are presented and two applications of free form volumes are discussed: definition of solid primitives and curve and surface modelling by the way of volume deformation.
... Bézier volumes are particular cases with the special coefficients c i, j,k = ( m i )( n j )( l k ) m m n n l l [6]. Definition 1 is equivalent to the traditional definition introduced in much of the literature [25][26][27]. When the lattice polytope is m,n,l , the toric surface will be a 3D Bézier volume. ...
... Therefore, 3D Bézier volumes have the same properties as toric surfaces. We will illustrate some of the properties as follows (see [25]): ...
The one-to-one property of injectivity is a crucial concept in computer-aided design, geometry, and graphics. The injectivity of curves (or surfaces or volumes) means that there is no self-intersection in the curves (or surfaces or volumes) and their images or deformation models. Bézier volumes are a special class of Bézier polytope in which the lattice polytope equals . Piecewise 3D Bézier volumes have a wide range of applications in deformation models, such as for face mesh deformation. The injectivity of 3D Bézier volumes means that there is no self-intersection. In this paper, we consider the injectivity conditions of 3D Bézier volumes from a geometric point of view. We prove that a 3D Bézier volume is injective for any positive weight if and only if its control points set is compatible. An algorithm for checking the injectivity of 3D Bézier volumes is proposed, and several explicit examples are presented.
... In this more general setting we shall define NURBS as usual but with two generalizations. Firstly we shall allow any number of parameter variables, using a tensor-product scheme similar to NURBS surfaces and volumes [23]. Secondly we will allow the control points to be in any vector space, not just the usual Euclidean two or three dimensional space. ...
... We can represent all of the above as generalized NURBS-generalized in the sense that there can be any number of parameters, not just curves and surfaces, and that the control points are not necessarily points in space. This allows us to represent for example volumes [23] and motions in SE(3) [20]. ...
A major aim of robotics research is the provision of systems which simplify the programming of robots, enabling experienced designers and engineers to implement robotic devices as part of a larger systems without the need to become expert programmers. Also in the quest for a flexible industrial production system is it desirable to be able to reprogram robots offline, so that they can be doing one task whilst being prepared for another. This paper describes the mathematical and computational background to a system which enables the development and testing of robot programs in a CAD environment in a way that is simple to use and compatible with existing methods used in CAD systems
... The extension to an AFFD of higher degree is a straightforward exercise. In particular, with a quadratic (degree n = 2) tetrahedral Bézier volume as deformation tool, at each plane we reproduce the quadratic construction of Section 3.2 (i.e., an inverse of a stereographic projection), so that planes transform to quadrics through O. Instead of a Bézier tetrahedron, we can employ a parallelepiped P as deformation tool, i.e., a trivariate tensor-product Bézier volume in rational version (Kalra et al., 1992;Davis and Burton, 1991;Lasser, 1994). Now, the user defines the reference P embedding the object, and then chooses the degree of the deformation tool. ...
In an optical anamorphosis, an object is seen distorted unless the viewer is positioned at a specific point, where the object appears normal. We describe how to endow a rational Free-Form Deformation with an anamorphic character in a simple manner, obtaining an AFFD (Anamorphic Free-Form Deformation). Given a (planar or 3D) initial object, which will appear normal from the desired viewpoint, we deform the object with a rational Bézier surface or volume as deformation tool. To achieve the desired deformation, the user input amounts to displacing the control points of the deformation tool along radial directions through the viewpoint, whereas the weights come as a byproduct. Mathematically, the deformation means changing the last homogeneous coordinate of the control points. An AFFD defined by a linear Bézier tetrahedron can be regarded as a user-friendly way to construct a perspective collineation. In this case, or when the deformation tool is a Bézier triangle of degree one, the AFFD transforms NURBS to NURBS keeping the original degree. With a deformation tool of higher degree, the rational composition required to obtain the exact result yields NURBS also of higher degree. For a quadratic Bézier triangle as deformation tool, our AFFD coincides with the inverse of a stereographic projection.
The task of modeling material heterogeneity (composition variation) is a critical issue in the design and fabrication of heterogeneous objects. Existing methods cannot efficiently model the material heterogeneity, due to the formidable size of the degrees of freedom for the specification of heterogeneous objects. In this research, we provide an intuitive way to model the object heterogeneity by using only a few parameters. These parameters carry physical meanings, such as diffusion coefficients in the diffusion process. We use a B-spline representation to model heterogeneous objects and material properties. We use diffusion equations to generate heterogeneous material composition profile. We then use finite element techniques to solve the material composition equations for the diffusion process. Finally we extend this method to the direct manipulation of material properties in heterogeneous objects.
Virtual reality( VR)provides a design space consisting of three-dimensional computer images where participants can interact with these images using natural human motions in real time. In the field of engineering design, prototyping and design verification have provided the initial application areas for VR. The research presented in this paper takes the scenario one step further by incorporating free-form deformation techniques and sensitivity analysis into the virtual world such that the designer can easily implement analysis-based shape design of a structural system where stress considerations are important. NURBS-based free-form deformation (NFFD) methods and direct manipulation techniques are used as the interface between the VR interaction and the finite element model. Sensitivity analysis is used to allow the designer to change the design model and immediately view the effects without performing a re-analysis. An engine connecting rod is analyzed to demonstrate how virtual reality techniques can be applied to structural shape design.
The paper demonstrates how NURBS-based Free-Form Deformations (NFFDs) can be used to generate and sculpt NURBS based objects in a way that avoids many of the limitations of ordinary FFDs. A system is described in which the user interacts with the NFFD through a set of direct acting tools, removing the need to manipulate the mesh directly. As the user applies the tools to the object, the NFFD is updated automatically. Objects are represented as isosurfaces of the NFFD volume, giving a good insight into the way an NFFD would deform a general object
The use of energy methods and variational principles is widespread in many fields of engineering of which structural mechanics and curve and surface design are two prominent examples. In principle many different types of function can be used as possible trial solutions to a given variational problem but where piecewise polynomial behaviour and user controlled cross segment continuity is either required or desirable, B-splines serve as a natural choice. Although there are many examples of the use of B-splines in such situations there is no common thread running through existing formulations that generalises from the one dimensional case through to two and three dimensions. We develop a unified approach to the representation of the minimisation equations for Bspline based functionals in tensor product form and apply these results to solving specific problems in geometric smoothing and finite element analysis using the Rayleigh-Ritz method. We focus on the development of algorithms for the exact computation of the minimisation matrices generated by finding stationary values of functionals involving integrals of squares and products of derivatives, and then use these to seek new variational based solutions to problems in the above fields. By using tensor notation we are able to generalise the methods and the algorithms from curves through to surfaces and volumes. The algorithms developed can be applied to other fields where a variational form of the problem exists and where such tensor product B-spline functions can be specified as potential solutions. ii Contents 1
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 [...
A methodology is proposed for creating and animating computer generated characters which combines recent research advances in robotics, physically based modeling and geometric modeling. The control points of geometric modeling deformations are constrained by an underlying articulated robotics skeleton. These deformations are tailored by the animator and act as a muscle layer to provide automatic squash and stretch behavior of the surface geometry. A hierarchy of composite deformations provides the animator with a multi-layered approach to defining both local and global transition of the character's shape. The muscle deformations determine the resulting geometric surface of the character. This approach provides independent representation of articulation from surface geometry, supports higher level motion control based on various computational models, as well as a consistent, uniform character representation which can be tuned and tweaked by the animator to meet very precise expressive qualities. A prototype system (Critter) currently under development demonstrates research results towards layered construction of deformable animated characters.
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
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.
A technique is presented for deforming solid geometric models in a free-form manner. The technique can be used with any solid modeling system, such as CSG or B-rep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces, for example. The deformation can be applied either globally or locally. Local deformations can be imposed with any desired degree of derivative continuity. It is also possible to deform a solid model in such a way that its volume is preserved.The scheme is based on trivariate Bernstein polynomials, and provides the designer with an intuitive appreciation for its effects.
Current research efforts focus on providing interactive techniques that make 3D concepts easy to use and accessible to large numbers of people. In this paper, a new interactive technique for animating deformable objects is presented. The technique allows easy specification and control of a class of deformations that cannot be produced by existing techniques without considerable human intervention.The methodology proposed relies on the Free-Form Deformation (FFD) technique. It makes use of a deformation tool that is animated like any other object.This approach provides a representation of the deformations independent of the surface geometry, and can be easily integrated into traditional hierarchical animation systems.
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.
Thesis (doctoral)--Technische Hochschule Darmstadt, 1991.
Current research efforts focus on providing more efficient and effective design methods for 3D modeling systems. In this paper a new deformation technique is presented. Among other things, arbitrarily shaped bump can be designed and surfaces can be bent along arbitrarily shaped curves. The purpose of this research is to define a highly interactive and intuitive modeling technique for designers and stylists. A natural way of thinking is to mimic traditional trades, such as sculpturing and moulding. Furthermore, with this deformation technique, the modeling tool paradigm is introduced. The object is deformed with a user-defined deformation tool. This method is an extension of the Free-Form Deformation (FFD) technique proposed by Sederberg and Parry.
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.
Definition of solid primitives by rational Bézier volumes, Interner Bericht #205/90
- D. Lasser
- P. Kirchgeßner
- D. Lasser
- P. Kirchgeßner
Rationale B~zier-volumina
- P Kirchgeflner
P. Kirchgeflner, Rationale B~zier-volumina, Diplomarbeit, Darmstadt, (1989).
Definition of solid primitives by rational B4zier volumes
- D Lasser
- P Kirchgeflner
D. Lasser and P. Kirchgeflner, Definition of solid primitives by rational B4zier volumes, Interner Bericht ##205/90, Fachbereich Informatik, Universit~t Kaiserslautern, (1990).
Definition of solid primitives by rational Bézier volumes, Interner Bericht #205/90
- Lasser