Geometric Modeling is an area drawing from computer science, mathematics, engineering, and the life sciences. It is concerned with the computer representation of objects as diverse as - brain scans - mathematical functions - terrains - airplane wings and many more. The seminar succeeded in bringing together leading researchers to present and discuss radically different approaches to the challenge of modeling complex geometric phenomena on the computer. Acquisition, representation and analysis of 3-dimensional geometry call for the combination of technically complex and often interdisciplinary approaches that are grounded both in classical mathematics and computer science data structures and theory.
From May 26 to May 30 2008 the Dagstuhl Seminar 08221 ``Geometric Modeling'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
Geometric invariants of triangular quadric patches Gudrun Albrecht University of Valenciennes, France We consider a rational triangular Bzier patch of degree 2. In general its underlying implicit surface is a so-called Steiner surface, an algebraic surface of order 4, which under certain conditions for the control points and the weights becomes a quadric surface. An easy method for determining, if the given patch lies on a quadric, and if so, for establishing the quadric’s affine type, has been presented in CAGD 15(1998). On this basis we consider the problem of determining the directions of the principal axes of the underlying quadric of the given patch as well as the position of the origin yielding the quadric’s normal form. Non-conforming shape representations Marc Alexa TU Darmstadt, Germany Non-conforming surface representations are based on a set of primitives scattered over the surface. These primitives could be points or low degree polynomial patches. We find this representation interesting, because it is often easier to generate from a given surface description and it might yield smaller Hausdorff error for the same number of primitives compared to continuous (i.e. connected) surface representations. We discuss two aspects of non-conforming surface representations: First, how to generate them based on spatial subdivision and, second, how to render them efficiently without conversion to a mesh. Splat representation of parametric surfaces D. Ayala Universitat Politècnica de Catalunya, Barcelona, Spain Point-based geometry representations and their splat-based generalisations have become a suitable technique both for modeling and rendering complex 3D shapes. So, it seems interesting to convert other kind of models to a point or splat-based representations.
From 29.05.05 to 03.06.05, the Dagstuhl Seminar 05221 ``Geometric Modeling'' was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. Abstracts of the presentations given during the seminar as well as abstracts of seminar results and ideas are put together in this paper. The first section describes the seminar topics and goals in general. Links to extended abstracts or full papers are provided, if available.
In this chapter the use of compactly-supported radial basis functions for surface reconstruction is described. To solve the problem of reconstruction or volume data generation specially designed software is employed. Time performance of the algorithm is investigated. Thanks to the efficient octree algorithm used in this study, the resulting matrix is a band diagonal matrix that reduces computational costs.
Declarative modeling is an emergent research domain in computer-aided geometric design. To deal with the generation problems in declarative modeling, a recent approach consists in using constraint satisfaction techniques. After an introduction to constraint satisfaction and an overview of related works in declarative modeling, this chapter presents an object-oriented constraint solver. This generic tool is based on constraint propagation and domain reduction and supports heterogeneous parameters. It can be used in many generation systems. As an application, a declarative modeler for virtual 3D-environments planning is briefly presented.
We describe an automatic surface reconstruction technique from a set of planar data points organised in parallel sections. The reconstruction employs skeletal implicit surfaces. Two key points in this study are: - Calculation of the 3D skeleton by establishing a correspondence between each pair of 2D Voronoi skeleton of two neighbouring sections. - Use of a uniform field function necessitating the introduction of the notion of "weighted skeleton".Another key point of this work is the proposition of an animation methodology by transforming the skeleton into a deformable mass/spring system.We also describe the multi layer particle system model, which is used for reconstruction and simulation of deformable objects. The last part of this work consists in defining an hybrid approach by using the two previous model in the same scene, improving the simulation.This work has been used in the context of a medical project to simulate the dynamic behaviour of organs during the conformal radiotherapy treatment.
Visualization of large geometric environments has always been an exciting project for computer graphics practitioners. Modern graphics workstations allow rendering of millions of polygons per second. Although these systems are impressive, they cannot catch up with the quality demanded by graphics systems used for visualizing complex geometric environments. After all, in such systems the amount of data that need to be processed increases dramatically as well. No matter how much graphics hardware evolves, it looks like practitioners are going to crave for what is impracticable for such hardware to render at interactive frame rates. In this chapter, we present some modeling techniques to overcome the problem of graphics hardware bottleneck in a particular context, viz. visualization of terrains and urban environments.
In this chapter we present algorithms for the representation and classification of the boundaries of two- and three-dimensional objects. The techniques covered are based on the Marching Cubes algorithm of Lorenson and Cline and differ from standard approximations such as those highlighted in Rosin as they do not involve any loss of data when using pixelised values.The techniques also permit a method for the charcterisation of objects by their boundaries; all that is required is the application of standard graph comparison algorithms on these modified data structures.
This chapter proposes a rational classification of Islamic Geometric Patterns (IGP) based on the Minimum Number of Grids (MNG) and Lowest Geometric Shape (LGS) used in the construction of the symmetric elements. The existing classification of repeating patterns by their symmetric groups is in many cases not appropriate or prudent [13]. The symmetry group theories do not relate to the way of thinking of the artisans involved, and completely has ignored the attributes of the unit pattern and has focused exclusively on arrangement formats. The chapter considers the current symmetric group theories only as arrangement patterns and not as classifications of IGP since they have a "global approach" and have failed to explore the possibilities in the construction elements of IGP. We describe and demonstrate procedures for constructing Star/Rosette unit patterns based on our proposed classification in a grid formation dictated by the final design of the unit pattern.
This chapter presents a new and direct approach for Loop subdivision surface fitting from a dense triangular mesh with arbitrary topology. The initial mesh model is first simplified with a topology-and feature-preserving mesh decimation algorithm. The simplified mesh is further used as the topological model of a Loop subdivision surface. The control vertices of the subdivision surface are finally fitted from a subset of vertices of the original dense mesh. During the fitting process, both the subdivision rules and position masks are used for setting up the observation equations. The emphasis of this chapter is on fitting issues. While only the Loop subdivision scheme is discussed in this chapter, the approach is applicable to any stationery subdivision scheme.
A scan-line algorithm uses a one-dimensional set operation instead of a depth comparison to find a visible surface of the object shape defined by constructive solid geometry (CSG). Interactive solid modelers employing a scan-line algorithm require considerable computing time in repeated construction and destruction of parts of the CSG object models during the design process. We discuss three steps to improve a scan-line algorithm to display the CSG models on a screen. First, changing the order of application of the partial procedures in the algorithm can improve the performance in the interactive modelers. Secondly, substituting set operations for simple manipulations of the elements in two kinds of lists reduces computing time. Finally, using the connection information at the adjoining facets drastically reduces the number of set operation executions.
Existing CAD systems are still monolithic and are based on complex geometrical models. Our aim is to implement a new approach, taking advantage of the possibilities of the World Wide Web and providing a novice end-user with an easy-to-use system. In order to get the same performance as a local system, it is necessary to provide our system with an adequate data representation and architecture. A study realised by our team has shown that classical models are based on information that is too low-level for our specifications and that a functional model is needed. This paper deals with our first results concerning the architecture which is to be the basis of different implementations, and it presents a first operational prototype. The first section introduces the tests we carried out in order to understand the advantages and drawbacks of different alternatives, and it describes the final choices for the architecture. The second section describes a prototype proving the feasibility of our approach. It is based on new intuitive interaction techniques, though we merely outline this point in this paper
This chapter presents a novel method for computing the intersection curves of two surfaces of revolution RSIC. In this method, each surface of revolution is decomposed into a collection of coaxial spherical stripes along the generatrix, by subdividing its generatrix into a collection of C0 or C1 coaxial circular arcs centered on the revolute axis. Thus, computing intersections of two surfaces of revolution RSIC is reduced to computing intersection curves of two spherical stripes SSIC. RSIC can be represented as a piecewise C0 or C1 circular approximation, which is quite convenient for various operations such as offsetting, blending and so on, To avoid the unnecessary intersection computations, cylindrical bounding shell CBS is devised and valid intersection intervals VII is introduced. Finally, a simple algorithm is designed to trace RSIC for classification.
This chapter introduces techniques used in muscle Modeling and rendering. An anatomically based approach to muscle Modeling is presented in which the muscle models are generated from anatomical data. These models provide a good visual description of muscle form and action and represent a sound base from which to progress further towards medically accurate simulation of human bodies. Deformation of these models is also performed on the basis of their anatomical structures. The result is an efficient, anatomically accurate, muscle representation that is specifically designed to accommodate the particular form of deformation exhibited by each individual muscle.Muscle rendering is addressed in two radically different ways. In the photorealistic approach, the muscle texture is generated from anatomical data, and a texture synthesis method is used to simulate the fibre patterns on the muscle surface. In the non-photorealistic method, the muscles are rendered directly from volumetric data using a pen-and-ink styles. This provides a useful alternative to traditional visualisation techniques.
Solving geometric metric constraints is a topical issue in CAD. An original way to solve a constraint system is to use geometric methods, providing a symbolic construction plan. Then, this plan can be numerically interpreted to generate the required figure. If multiple solutions are produced, most solvers propose to scan the entire space of the solutions found, that is generally tedious. We show how the inner properties of a symbolic solver allow to deal more efficiently with this case. After briefly recalling our sketch-based selection method, that enables to easily eliminate most of the solutions and to keep the only, or at worst the few solutions that have the best likeness with the original drawing, we introduce a new step by step interpretation mechanism implemented as a debugger-like tool, that allows to browse the remaining solutions tree in order to help the user choosing the required solution
Surface representation using the solution to a partial differential equation (PDE) is an important topic of computer graphics and computer-aided design. In existing references, various free-from surfaces were created with a 4th-order PDE, which is only able to meet the tangential conditions at the surface boundaries. The need for a 6th-order PDE in surface modelling arises in two situations: one is to generate surfaces with curvature continuity and the other is to use curvature values as a user handle for surface shape manipulation. In this paper, we introduce such a 6th-order PDE for free-form surface generation and develop a finite difference method to solve this PDE. We also investigate the effects of boundary curvature and the vector-valued shape parameters on the surface shape. It was found that their variation has a strong influence on the shape of the surfaces and that, therefore, they can be used as flexible user handles
Generally, in the field of industrial design, measurement data of a clay model are faired, and its three dimensional model is made in CAD, However, it is a problem that this fairing work take a long time and require a great deal of labor. We propose an automatic fairing system of a curve for the application to reverse engineering technology. The system is constructed by the following five steps. 1) Key-line data are extracted from a clay model by a 3D-digitizer. 2) The extracted key-line data are approximated by a spline with a genetic algorithm. 3) The spline curve (key-line) is divided into curves of monotone curvature. 4) "Characteristic" of the curve is analyzed for each curve of monotone curvature, 5) Each curve of monotone curvature is substituted by a visual language based on the result of the analysis mentioned above, and each key-line is reconstructed by the visual language. By this system, we can easily obtain an aesthetic curve intended by a designer in a short time.
A smooth curve interpolation scheme for convex data has been developed. This scheme uses piecewise rational cubic spline functions with both unit and chord-length parametrization test. The necessary and sufficient conditions on inflection points and singularities for a convex data have been presented. Shape preserving parameters are automatically generated. The degree of smoothness attained is C2 which is more powerful than a previous C1 method.
Presents a method to connect VRML (Virtual Reality Modeling Language) and Java components in a Web page using EAI (External Authoring Interface), which makes it possible to interactively generate and edit VRML meshes. The meshes used are based on regular grids, to provide an interaction and modeling approach that uses the internal semantics of such a mesh by linking the available modeling operations to either single vertices, the vertices of a ring or column, or all the vertices from the VRML mesh. Our method permits strict mesh complexity control and scales the operations according to the mesh properties. We describe the structure of the system and provide a few examples of the meshes that were created.
