629 research outputs found
Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions
AbstractAlgorithms are developed, based on topological principles, to evaluate the boundary and “internal structure” of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the “internal” envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in R3, and other forms of geometrical convolution, are briefly discussed
Solid Modeling
To appear in the Encyclopedia of Electrical and Electronics Engineering, Ed. J. Webster, John Wiley & Sons, 1999.A solid model is a digital representation of the geometry of an existing or envisioned physical object. Solid models are used in many industries, from entertainment to health care. They play a major role in the discrete-part manufacturing industries, where precise models of parts and assemblies are created using solid modeling software or more general computer-aided design (CAD) systems. Solid modeling is an interdisciplinary field that involves a growing number of areas. Its objectives evolved from a deep understanding of the practices and requirements of the targeted application domains. Its formulation and rigor are based on mathematical foundations derived from general and algebraic topology, and from Euclidean, differential, and algebraic geometry. The computational aspects of solid modeling deal with efficient data structures and algorithms, and benefit from recent developments in the field of computational geometry. Efficient processing is essential, because the complexity of industrial models is growing faster than the performance of commercial workstations. Techniques for modeling and analyzing surfaces and for computing their intersections are important in solid modeling. This area of research, sometimes called computer aided geometric design, has strong ties with numerical analysis and differential geometry. Graphic user-interface (GUI) techniques also play a crucial role in solid modeling, since they determine the overall usability of the modeler and impace the user's productivity. There have always been strong symbiotic links and overlaps between the solid modeling community and the computer graphics community. Solid modeling interfaces are based on efficient three-dimensional (3D) graphics techniques, whereas research in 3D graphics focuses on fast or photo-realistic rendering of complex scenes, often composed of solid models, and on realistic or artistic animations of non-rigid objects. A similar symbiotic relation with computer vision is regaining popularity, as many research efforts in vision are model-based and attempt to extract 3D models from images or video sequences of existing parts or scenes. These efforts are particularly important for solid modeling, because the cost of manually designing solid models of existing objects or scenes far excees the other costs (hardware, software, maintenance, and training) associated with solid modeling. Finally, the growing complexity of solid models and the growing need for collaboration, reusability of design, and interoperability of software require expertise in distributed databases, constraint management systems, optimization techniques, object linking standards, and internet protocols. This report provides a brief overview of the solid modeling field, its fundamental technologies, and some important applications
Minkowski Sum Construction and other Applications of Arrangements of Geodesic Arcs on the Sphere
We present two exact implementations of efficient output-sensitive algorithms
that compute Minkowski sums of two convex polyhedra in 3D. We do not assume
general position. Namely, we handle degenerate input, and produce exact
results. We provide a tight bound on the exact maximum complexity of Minkowski
sums of polytopes in 3D in terms of the number of facets of the summand
polytopes. The algorithms employ variants of a data structure that represents
arrangements embedded on two-dimensional parametric surfaces in 3D, and they
make use of many operations applied to arrangements in these representations.
We have developed software components that support the arrangement
data-structure variants and the operations applied to them. These software
components are generic, as they can be instantiated with any number type.
However, our algorithms require only (exact) rational arithmetic. These
software components together with exact rational-arithmetic enable a robust,
efficient, and elegant implementation of the Minkowski-sum constructions and
the related applications. These software components are provided through a
package of the Computational Geometry Algorithm Library (CGAL) called
Arrangement_on_surface_2. We also present exact implementations of other
applications that exploit arrangements of arcs of great circles embedded on the
sphere. We use them as basic blocks in an exact implementation of an efficient
algorithm that partitions an assembly of polyhedra in 3D with two hands using
infinite translations. This application distinctly shows the importance of
exact computation, as imprecise computation might result with dismissal of
valid partitioning-motions.Comment: A Ph.D. thesis carried out at the Tel-Aviv university. 134 pages
long. The advisor was Prof. Dan Halperi
Quasiconvex Programming
We define quasiconvex programming, a form of generalized linear programming
in which one seeks the point minimizing the pointwise maximum of a collection
of quasiconvex functions. We survey algorithms for solving quasiconvex programs
either numerically or via generalizations of the dual simplex method from
linear programming, and describe varied applications of this geometric
optimization technique in meshing, scientific computation, information
visualization, automated algorithm analysis, and robust statistics.Comment: 33 pages, 14 figure
- …