Subdivision Tree Representation of Arbitrary Triangle Meshes

Computer Aided Geometric Design, 1986

This paper establishes a number of mathematical results relevant to the problem of constructing a triangulation, i.e., a simplical tessellation of the convex hull of an arbitrary finite set of points in n-space.

2000

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.

2008

Some basic mathematical tools such as convex sets, polytopes and combinatorial topology, are used quite heavily in applied fields such as geometric modeling, meshing, computer vision, medical imaging and robotics. This report may be viewed as a tutorial and a set of notes on convex sets, polytopes, polyhedra, combinatorial topology, Voronoi Diagrams and Delaunay Triangulations. It is intended for a broad audience of mathematically inclined readers. I have included a rather thorough treatment of the equivalence of V-polytopes and H-polytopes and also of the equivalence of V-polyhedra and H-polyhedra, which is a bit harder. In particular, the Fourier-Motzkin elimination method (a version of Gaussian elimination for inequalities) is discussed in some detail. I also included some material on projective spaces, projective maps and polar duality w.r.t. a nondegenerate quadric in order to define a suitable notion of ``projective polyhedron'' based on cones. To the best of our knowledge, this notion of projective polyhedron is new. We also believe that some of our proofs establishing the equivalence of V-polyhedra and H-polyhedra are new.

Handbook of Computational Geometry, 2000

ISPRS International Journal of Geo-Information

In spatial science, the relationship between spatial objects is considered to be a vital element. Currently, 3D objects are often used for visual aids, improving human insight, spatial observations, and spatial planning. This scenario involves 3D geometrical data handling without the need for topological information. Nevertheless, in the near future, users will shift to more complex queries corresponding to the existing 2D spatial approaches. Therefore, having 3D spatial objects without having these relationships or topology is impractical for 3D spatial analysis queries. In this paper, we present a new method for creating topological information that we call the Compact Cell Complexes (CACC) data structure for 3D spatial objects. The idea is to express in the most compact way the topology of a model in 3D (or more generally in nD) without requiring the topological space to be discrete or geometric. This is achieved by storing all the atomic cycles through the models (null combinato...

2000

This work introduces a scalable topological data structure for manifold triangular meshes called Compact Half-Edge (CHE). It provides a high degree of scalability, since it is able to optimize the memory consumption / execution time ratio for different applications and data by using features of its different levels. An object-oriented API using class inheritance and virtual instantiation enables a unique

2008

Recent advances in acquisition and modelling techniques led to generating an exponentially increasing amount of 3D shapes available both over the Internet or in specific databases. While the number grows it becomes more and more difficult to keep an organized knowledge over the content of this repositories. It is commonly intended that in the near future 3D shapes and models will be indexed and searched using procedure and instruments mimicking the same operations performed on images while using algorithms, data structures and instruments peculiar to the domain. In this context it is thus important to have tools for automatic characterization of 3D shapes, and skeletons and partitions are the two most prominent ones among them. In this paper we will describe an experience of building some of this tools on the top of a popular and robust library for manipulating meshes (OpenMesh). The preliminary results we present are promising enough to let us expect that the sum of the tools will be a useful aid to improving indexing and retrieval of digital 3D objects. The work presented here is part of a larger project: Three-Dimensional Shape Indexing and Retrieval Techniques (3-SHIRT), in collaboration with the Universities of Genoa, Padua, Udine, and Verona.

Discrete & Computational Geometry, 2010

Preserving topological properties of objects during reduction procedures is an important issue in the field of discrete image analysis. Such procedures are generally based on the notion of simple point, the exclusive use of which may result in the appearance of "topological artifacts." This limitation leads to consider a more general category of objects, the simple sets, which also enable topology-preserving image reduction. A study of two-dimensional simple sets in two-dimensional spaces has been proposed recently. This article is devoted to the study of two-dimensional simple sets in spaces of higher dimension (i.e., n-dimensional spaces, n ≥ 3). In particular, several properties of minimal simple sets (i.e., which do not strictly include any other simple sets) are proposed, leading to a characterisation theorem. It is also proved that the removal of a two-dimensional simple set from an object can be performed by only considering the minimal ones, thus authorising the development of efficient thinning algorithms.

We introduce HPS, a new topological data structure that efficiently represents hierarchies of planar subdivisions, thus providing direct and efficient support for GIS concepts such as abstract generalizations and multi-scale partitions. Unlike previous ad hoc solutions, HPS provides efficient access to adjacency information for each level and across levels, while storing the complete hierarchy in a single data structure, without duplications. HPS also provides topological operators that ensure global consistency. Like all topological data structures, HPS can be used as a framework onto which geometric and attribute information is placed: HPS explicitly handles attributes consistently with modeling, and naturally supports both topological and geometrical multi-resolution representations. We also discuss how some typical applications in GIS, Digital Cartography, and Finite Element mesh generation can be improved with HPS. keywords: topological data structures, hierarchical mode...

IEEE Transactions on Pattern Analysis and Machine Intelligence, 2000

A point of a discrete object is called simple if it can be deleted from this object without altering topology. In this article, we present new characterizations of simple points which hold in dimensions 2, 3 and 4, and which lead to efficient algorithms for detecting such points. In order to prove these characterizations, we establish two confluence properties of the collapse operation which hold in the neighborhood of a point in spaces of low dimension. This work is settled in the framework of cubical complexes, which provides a sound topological basis for image analysis, and allows to retrieve the main notions and results of digital topology, in particular the notion of simple point.