Abel J. P. Gomes

Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms

Implicit objects have gained increasing importance in geometric modeling, visualisation, animation, and computer graphics, because their geometric properties provide a good alternative to traditional parametric objects. This book presents the mathematics, computational methods and data structures, as well as the algorithms needed to render implicit curves and surfaces, and shows how implicit objects can easily describe smooth, intricate, and articulatable shapes, and hence why they are being increasingly used in graphical applications. Divided into two parts, the first introduces the mathematics of implicit curves and surfaces, as well as the data structures suited to store their sampled or discrete approximations, and the second deals with different computational methods for sampling implicit curves and surfaces, with particular reference to how these are applied to functions in 2D and 3D spaces.

Form feature modelling in a hybrid CSG/BRep scheme

Abstract A new hybrid pseudo-CSG/BRep schema for product modelling is described. In this schema, there exists a clear distinction between morphology and the geometry of an object. At the highest level of abstraction, the object morphology or shape is hierarchically defined by means of a graph, called Feature Adjacency Graph (FAG), in which the nodes are either positive or negative volumetric cells (form features) and the arcs are adjacency/interaction relationships between those cells. The hierarchical morphological characterization of an object made by the FAG allows to one decrease or even suppress the semantic gap between the product model and the underlying solid model since each cell always embodies a form feature. In this manner it becomes possible to capture the designer intent, as well as to help the CAD/CAM integration. At the low level of abstraction, i.e., the BRep level, two associated data structures are maintained. The first, called Feature Topological Structure (FTS), holds the family of volumetric cells, each one of them represented explicitly by its surface boundary (BRep). The second constitutes the evaluated BRep model itself for the object solid. It should be noted that any constructive or destructive compositional action in the object definition occurs simultaneously in both data structures. In a way, the FTS retains at a low level, the construction history of the object model. Besides, the availability of the topological entities belonging to the BRep allows a graphical dialog with the model and also facilitates the generation of instructions for NC milling machines. In short, this hybrid pseudo-CSG/BRep intends to capture the virtues of both representation schemas and eliminate as much as possible as their underlying drawbacks.

CUDA-based triangulations of convolution molecular surfaces

Computing molecular surfaces is important to measure areas and volumes of molecules, as well as to infer useful information about interactions with other molecules. Over the years many algorithms have been developed to triangulate and to render molecular surfaces. However, triangulation algorithms usually are very expensive in terms of memory storage and time performance, and thus far from real-time performance. Fortunately, the massive computational power of the new generation of low-cost GPUs opens up an opportunity window to solve these problems: real-time performance and cheap computing commodities. This paper just presents a GPU-based algorithm to speed up the triangulation and rendering of molecular surfaces using CUDA. Our triangulation algorithm for molecular surfaces is based on a multi-threaded, parallel version of the Marching Cubes (MC) algorithm. However, the input of our algorithm is not the volume dataset of a given molecule as usual for Marching Cubes, but the atom centers provided by the PDB file of such a molecule. We also carry out a study that compares a serial version (CPU) and a parallel version (GPU) of the MC algorithm in triangulating molecular surfaces as a way to understand how real-time rendering of molecular surfaces can be achieved in the future.

A Derivative-Free Tracking Algorithm for Implicit Curves with Singularities

This paper introduces a new algorithm for rendering implicit curves. It is curvature-adaptive. But, unlike most curve algorithms, no differentiation techniques are used to compute self-intersections and other singularities (e.g. corners and cusps). Also, of theoretical interest, it uses a new numerical method for sampling curves pointwise.

A mathematical model for boundary representations of n -dimensional geometric objects

The major purpose of this paper is to introduce a general theory within which previous boundary representations (Breps) are a special case. Basically, this theory combines sub-analyt,ic geometry and theory of stratifications. The sub-analyt,ic geometry covers almost, all geometric engineering artefacts, and it is a generalisation of the semi-analytic geometry, which in turn is a generalisation of the semialgebraic geometry used by most geometric kernels. On the other hand, the theory of stratifications provides the most general manifold structures for geometric objects that it is possible to consider in geometric modelling. Whitney stratifications are particularly useful in geometric modelling because they provide a general abstraction for the #structure of boundary representations of objects in IV’. Remarkably, it is well-known in mathematics that sub-analytic objects are Whitney stratifiable, and this mathematically matches and validates the usual geometry-structure design of boundary representation data structures. Thus, the general B-rep introduced here represents Whitney-stratified sub-analytic objects, though the global design of the data structure is classical: the geometry (sub-analytic geometry) separated from the structure (Whitney stratification). 1 Theoretical evolution of boundary repre-

Set-combinations of the mixed-dimension cellular objects of the Djinn API

Abstract This paper is concerned with the mathematics and formal specification of “set-like” operations for the mixed dimension cellular geometric objects of the Djinn Application Programming Interface. The relationships between these operations and stratifications of dimensionally heterogeneous semianalytic point-sets are uncovered and formalised. Semianalytic geometry is central to the algebraic model discussed in this paper, but multi-disciplinary concepts from topology, differential geometry and computer-aided geometric design have been used also. In particular, the use of strong relative topological stratifications enables Djinn to satisfy significant industrial requirements.

Graphics processing unit-based triangulations of Blinn molecular surfaces

Computing the surface of a molecule (e.g., a protein) plays an important role in the analysis of its geometric structure as needed in the study of interactions between proteins, protein folding, protein docking, and so forth. There are a number of algorithms for the computation of molecular surfaces and their triangulations, but only a few take advantage of graphics processing unit computing. This paper describes a graphics processing unit‐based marching cubes algorithm to triangulate molecular surfaces. In the end of the paper, a performance analysis of three implementations (i.e., serial CPU, CUDA, and OpenCL) of the marching cubes‐based triangulation algorithm takes place as a way to realize beforehand how molecular surfaces can be rendered in real‐time in the future. Copyright

Normal-based simplification algorithm for meshes

This paper presents a new edge collapsing-based simplification algorithm for meshes. It is based on the variation of the vectors that are normal to faces around the collapsing edge. Its main novelty is that it uses the same criterion to choose and validate the collapsing edge. Besides, at the best knowledge of the authors, it is the fastest simplification algorithm found in the literature using the edge collapse operation. This simplification algorithm, in conjunction with its inverse algorithm (i.e. refinement algorithm), allows the automatic creation of a multiresolution schema, i.e. a sequence of meshes at different resolutions. Additionally, it makes a good trade-off between time performance and mesh quality

POINT-SETS AND CELL STRUCTURES RELEVANT TO COMPUTER AIDED DESIGN

Several fields of mathematics are relevant to computer aided design and other software systems involving solid object geometry, topology, differential and algebraic geometry being particularly important. This paper discusses some of this mathematics in order to provide a theoretical foundation for geometric modelling kernels that support non-manifold objects with an internal cellular structure and subsets of different dimensions. The paper shows relationships between relevant concepts from topology, differential geometry and computer aided geometric design that are not widely known in the CAD community. It also discusses semialgebraic, semianalytic and subanalytic sets as candidates for object representation. Stratifications of such sets are proposed for an objects cellular structure and new stratification concepts are introduced to support candidate applications.

A CUDA-Based Implementation of Stable Fluids in 3D with Internal and Moving Boundaries

Fluid simulation has been an active research field in computer graphics for the last 30 years. Stams stable fluids method, among others, is used for solving the equations that govern fluids (i.e. Navier-Stokes equations). An implementation of stable fluids in 3D using NVIDIA Compute Unified Architecture, shortly CUDA, is provided in this paper. This CUDA-based implementation also features the accurate physical treatment of internal (i.e. static boundaries inside the simulation domain) and moving boundaries. The performance gains of the presented implementation vs a sequential CPU-based implementation, and points of further improvement are also addressed.