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Topic: Combinatorial geometry

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	Computational geometry - Wikipedia, the free encyclopedia
		In computer science, computational geometry is the study of algorithms to solve problems stated in terms of geometry.
		The main impetus for the development of computational geometry as a discipline was progress in computer graphics, computer-aided design and manufacturing (CAD/CAM), but many problems in computational geometry are classical in nature.
		The primary goal of research in combinatorial computational geometry is to develop efficient algorithms and data structures for solving problems stated in terms of basic geometrical objects: points, line segments, polygons, polyhedra, etc.
	en.wikipedia.org /wiki/Computational_geometry (640 words)

	Discrete geometry - Wikipedia, the free encyclopedia
		Discrete geometry or combinatorial geometry may be loosely defined as study of geometrical objects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely on the notion of continuity.
		Parts of its domain of research is often attributed to other kinds of geometry: digital geometry, computational geometry.
		Kepler's conjecture (Johannes Kepler, 1611): The densest way to pack identical spheres in a given space is the "cannonball" arrangement, i.e., in flat layers, with each sphere resting upon three touching spheres beneath it.
	en.wikipedia.org /wiki/Combinatorial_geometry (154 words)

	Mathematics - Wikipedia, the free encyclopedia
		In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations), to the empirical mathematics of the various sciences (applied mathematics), and more recently to the rigorous study of uncertainty.
		Within differential geometry are the concepts of fiber bundles and calculus on manifolds.
		Within algebraic geometry is the description of geometric objects as solution sets of polynomial equations, combining the concepts of quantity and space, and also the study of topological groups, which combine structure and space.
	en.wikipedia.org /wiki/Mathematics (3891 words)

	Combinatorial Solid Geometry, Boundary Representations, and Non-Manifold Geometry
		For example, the geometry information associated with a planar face is the plane equation which includes the outward-pointing surface normal; the plane equation does not have to be re-derived from the vertices.
		Therefore, the conclusion was that the existing combinatorial solid geometry database would remain unchanged, and that any application that required an explicit representation of the modeled shapes would obtain that explicit representation through some form of conversion of the underlying CSG database.
		Within the geometry editor {\bf mged}, a high-speed rendering of a polygonal approximation of a shape is very useful for inspecting the model.
	ftp.arl.mil /~mike/papers/90nmg/joined.html (20066 words)

	FLUKA online manual, 2005.6
		The Combinatorial Geometry (CG) used by FLUKA is a modification of the package developed at ORNL for the neutron and gamma-ray transport program MORSE [Emm75] which was based on the original combinatorial geometry by MAGI (Mathematical Applications Group, Inc.) [Gub67,Lic79].
		A single geometry can be a mixture of modular areas, described by lattices, and "normal" areas, described by standard regions.
		Geometry debugger: The GEOEND card can be used also to activate the geometry debugger, using the WHAT and SDUM parameters.
	www.fluka.org /manual/sect/s094/text.html (4842 words)

	Combinatorial geometry (Site not responding. Last check: 2007-10-31)
		Discrete geometry or combinatorial geometry may be loosely defined as study of geometricalobjects and properties that are discrete or combinatorial, either by their nature or by their representation; the study that does not essentially rely onthe notion of continuity.
		Parts of its domain of research is often attributed to other kinds of geometry: digital geometry, computationalgeometry.
		Kepler's conjecture (Johannes Kepler, 1611): The densest way to pack identical spheres in a given space is the "cannonball"arrangement, i.e., in flat layers, with each sphere resting upon three touching spheres beneath it.
	www.therfcc.org /combinatorial-geometry-211090.html (115 words)

	Geometry Algorithm Textbooks
		This book presents a systematic treatment of the foundations of computational geometry and presents rigorous algorithmic solutions that are efficient in practical situations.
		Although it seems specialized, this method is currently a successful practical approach for implementing many geometry algorithms, including: computing trapezoidal decompositions, Voronoi diagrams, point location and range queries in arrangements of hyperplanes, constructing convex polytopes, and hidden surface removal.
		This is a polished advanced text about computational geometry algorithms and their connection with the older field of combinatorial geometry.
	www.geometryalgorithms.com /books_textbooks.shtml (838 words)

	ScienceDaily: Mathematics (Site not responding. Last check: 2007-10-31)
		In addition to these main concerns, there are also subdivisions dedicated to exploring links from the heart of mathematics to other fields: to logic, to set theory (foundations) and to the empirical mathematics of the various sciences (applied mathematics).
		Algebraic geometry – Differential topology – Algebraic topology – Linear algebra – Combinatorial geometry – Manifolds
		In the case of geometry, for example, it is not relevant to mathematics to know whether points and lines exist in any physical sense, as geometry starts from axioms and postulates about abstract entities called "points" and "lines" that we feed into the system.
	www.sciencedaily.com /encyclopedia/mathematics (4227 words)

	Wiley::Combinatorial Geometry
		A complete, self-contained introduction to a powerful and resurging mathematical discipline … Combinatorial Geometry presents and explains with complete proofs some of the most important results and methods of this relatively young mathematical discipline, started by Minkowski, Fejes Tóth, Rogers, and Erd???s.
		Combinatorial Geometry will be of particular interest to mathematicians, computer scientists, physicists, and materials scientists interested in computational geometry, robotics, scene analysis, and computer-aided design.
		It is also a superb textbook, complete with end-of-chapter problems and hints to their solutions that help students clarify their understanding and test their mastery of the material.
	www.wiley.com /WileyCDA/WileyTitle/productCd-0471588903.html (173 words)

	Geometry and Topology, Department of Mathematics, UIUC
		The document Graduate Study in Geometry and Topology outlines the general areas of geometry and topology studied here and describes the advanced undergraduate and graduate courses that are offered regularly.
		Combinatorial group theory, decision problems, automata theory and formal language theory, computational complexity.
		Differential geometry, foliation theory, gauge theory, moduli spaces, low dimensional geometry and topology, topological quantum field theory.
	www.math.uiuc.edu /GraduateProgram/researchmath/geomtop.html (342 words)

	Combinatorial Geometry in Characteristic 1 - Borovik, Gelfand, White (ResearchIndex)
		Abstract: Introduction Many geometries over fields have formal analogues which can be thought of as geometries over the field of 1 element 1.
		There are indications that combinatorial flag varieties are very suitable for cohomological calculations.
		Borovik, I. Gelfand and N. White, Combinatorial geometry in characteristic 1, Manchester Centre for Pure Mathematics, preprint 1999/10, 30 pp.
	citeseer.ist.psu.edu /487035.html (919 words)

	Northeastern University, Department of Mathematics
		Discrete and combinatorial geometry is mainly represented by Egon Schulte.
		He studies discrete geometric structures such as polytopes, polyhedra, complexes, and tilings, as well as their geometric, combinatorial, and algebraic symmetries.
		For instance, his study of multivariate discriminants and resultants in computational algebraic geometry (joint with I. Gelfand and M. Kapranov) led him to the discovery of an important new class of convex polytopes called secondary polytopes.
	www.math.neu.edu /research/combinsurvey.html (205 words)

	Amazon.com: Combinatorial Convexity and Algebraic Geometry (Graduate Texts in Mathematics): Books: Gunter Ewald,G]nter ... (Site not responding. Last check: 2007-10-31)
		This text provides an introduction to the theory of convex polytopes and polyhedral sets, to algebraic geometry and to the fascinating connections between these fields: the theory of toric varieties (or torus embeddings).
		This part of the book can thus serve for a one-semester introduction to algebraic geometry, with the first part serving as a reference for combinatorial geometry.
		Given a variety X, this procedure asks for a map from a nonsingular variety Y to X, such that the map is an isomorphism over the nonsingular locus of X. It was the case of a plane curve singularity that was essentially solved by Newton.
	www.amazon.com /exec/obidos/tg/detail/-/0387947558?v=glance (1088 words)

	TCS/DM: Essay on Discrete Mathematics
		The tight connection between Discrete Mathematics and Theoretical Computer Science, and the rapid development of the latter in recent years, led to an increased interest in Combinatorial techniques and to an impressive development of the subject.
		While many of the basic combinatorial results were obtained mainly by ingenuity and detailed reasoning, without relying on many deep, well developed tools, the modern theory has already grown out of this early stage.
		Combinatorial topics such as Ramsey Theory, Combinatorial Set Theory, Matroid Theory, Extremal Graph Theory, Combinatorial Geometry and Discrepancy Theory are related to a large part of the mathematical and scientific world, and these topics have already found numerous applications in other fields.
	www.math.ias.edu /csdm/dm.html (604 words)

	Project-Team-VEGAS:Combinatorial line geometry
		We obtained two new results on the theme of combinatorial line geometry.
		We improved the existing bound on the number of geometric permutations of disjoint balls with equal radii in any dimension [13] – a geometric permutation is an ordering that can be realized by a line intersecting all the balls.
		We also improved the known bounds on Helly numbers for the existence of a line intersecting a collection of disjoint balls of equal radii in 3 dimensions [15].
	www.inria.fr /rapportsactivite/RA2005/vegas/uid17.html (94 words)

	Joseph Malkevitch: Geometry in Utopia
		In light of this, and the fact that unlike many other parts of mathematics, geometry often requires relatively little background to get started, it sad how little awareness there is either within the mathematics community and certainly outside of it of the vast domains of geometry.
		Geometry can be thought of as being built up from many parts (not all incorporated below) even though these parts overlap greatly.
		Hadwiger, H. and H. Debrunner, V. Klee, Combinatorial Geometry in the Plane, Holt, Rhinehart, and Winston, New York, 1964.
	www.york.cuny.edu /~malk/utopia.html (5458 words)

	Combinatorial topology (Site not responding. Last check: 2007-10-31)
		In mathematics combinatorial topology was an older name for algebraic topology dating from the time when topological of spaces (for example the Betti numbers) were regarded as derived from combinatorial such as simplicial complexes.
		This point of view is often to Emmy Noether and so the change of title reflect her influence.
		Historically, combinatorial topology was a precursor to what is now the field of algebraic topology, and this book gives an elementary introduction to the subject, directed towards the beginning student of topology or geometry.
	www.freeglossary.com /Combinatorial_topology (282 words)

	Combinatorial and Computational Geometry - Cambridge University Press
		During the past few decades, the gradual merger of Discrete Geometry and the newer discipline of Computational Geometry has provided enormous impetus to mathematicians and computer scientists interested in geometric problems.
		It includes surveys and research articles exploring geometric arrangements, polytopes, packing, covering, discrete convexity, geometric algorithms and their complexity, and the combinatorial complexity of geometric objects, particularly in low dimension.
		There are points of contact with many applied areas such as mathematical programming, visibility problems, kinetic data structures, and biochemistry, as well as with algebraic topology, geometric probability, real algebraic geometry, and combinatorics.
	www.cambridge.org /catalogue/catalogue.asp?isbn=0521848628 (549 words)

	The Geometry Junkyard: Combinatorial Geometry
		What I mean by "combinatorial geometry" consists of problems in which one starts with a geometric figure (say a polytope) but then considers abstract incidence properties of it rather than its metric properties.
		Matthias Weber illustrates a three-dimensional generalization of Brianchon's theorem that the three long diagonals of a hexagon inscribed in a conic meet at a point.
		A Geometrical Picturebook of finite and combinatorial geometries, B.
	www.ics.uci.edu /~eppstein/junkyard/combinatorial.html (1682 words)

	SoCG 2000 -- SCG2000: Call for Papers
		The Sixteenth Annual Symposium on Computational Geometry, featuring an applied track, a theoretical track, and a video review, will be held at Hong Kong University of Science and Technology, Hong Kong.
		Topics for the theoretical track include, but are not limited to design and theoretical analysis of geometric algorithms and data structures; lower bounds for geometric problems; and discrete and combinatorial geometry.
		Topics for the applied track include, but are not limited to experimental analysis of algorithms and data structures; mathematical and numerical issues arising from implementations; and novel uses of computational geometry in other disciplines, such as robotics, computer graphics, geometric and solid modeling, manufacturing, and geographical information systems.
	www.cs.ust.hk /tcsc/scg00_5.html (534 words)

	Tom's Combinatorial Geometry Class (Site not responding. Last check: 2007-10-31)
		This page was developed in the Spring of 2000 for the course Combinatorial Geometry at Merrimack College.
		Convex Polyhedral Geometry: construction of paper models, Euler's Formula, planar duality, coloring theorems, Hamilton cycles, Buckyball classification and edge coloring, spherical geometry, solid angles, Descartes' Theorem, surfaces of higher genus, convex polytopes, simplices, the Generalized Euler's Formula.
		Combinatorial Modelling of Paper Folding: Maekawa's Theorem, Kawasaki's Theorem, local and global conditions for flat-foldability, NP-completeness, counting foldings, 3D solid paper folding, twists and origami tessellations, the Rabbit-Ear Theorem and origami design, isometries of the plane, high-dimensional flat folding.
	web.merrimack.edu /hullt/combgeom/combgeom.html (254 words)

	Info about Jesus Antonio de Loera
		This project is closely related to the understanding of the space of subdivisions of a convex polytope.
		Jesús holds a joint appointment as a Postdoctoral Research Fellow at the Geometry Center and as a visiting faculty member of the School of Mathematics at the University of Minnesota.
		Combinatorial models of real algebraic varieties: using computers in their topological classification (invited speaker) Department of Mathematics and Computer Science, Emory University, March (1997).
	www.geom.umn.edu /about/people/home/deloera.html (621 words)

	History of the Dept. (Site not responding. Last check: 2007-10-31)
		The chairman was György Hajós (geometry of numbers, foundation of geometry) and among the professors was László Fuchs (algebra, geometric groups).
		The chairman was József Molnár (combinatorial geometry, discrete geometry, foundation of geometry) and among the professors was Gyula Soós (differential geometry).
		The chairman was Károly Böröczky (combinatorial geometry, discrete geometry) and among his colleague professors was János Szenthe (differential geometry).
	www.cs.elte.hu /geometry/history.html (337 words)

	DIMACS Workshop on Pseudorandomness and Explicit Combinatorial Constructions (Site not responding. Last check: 2007-10-31)
		The development of the probabilistic method in combinatorics has led to a variety of non-constructive proofs of existence of combinatorial objects satisfying certain specified properties: graphs whose clique and independence numbers are both very small (Ramsey graphs), graphs with strong connectivity properties, such as expanders, and large codes with optimal distances.
		This symbiosis is also evident in the connection between the theory of derandomization in computational geometry and discrepancy theory within combinatorial geometry.
		In recent years, these techniques have included methods from such diverse areas as representation theory and algebraic geometry, and one aim of the workshop is to expose a broader range of researchers to these techniques.
	dimacs.rutgers.edu /Workshops/Pseudorandom/announcement.html (370 words)

	Citations: Combinatorial Geometry - Pach, Agarwal (ResearchIndex)
		Pach and P.K. Agarwal, Combinatorial Geometry, Wiley, New York, 1995.
		Pach and P.K. Agarwal, Combinatorial Geometry, John Wiley and Sons, New York, 1995.
		Clearly, the underlying abstract graph, G 0, whose vertex set is V (G) and whose edge set consists of those pairs of vertices which are connected in G by a segment, is planar.
	citeseer.ist.psu.edu /context/56947/0 (3777 words)

	Discrete and Combinatorial Geometryu (Site not responding. Last check: 2007-10-31)
		Discrete and Combinatorial Geometry is one of the fields that benefited most from this source.
		Discrete Mathematics is the associated field acting as a general framework, and most specific problems on foundations investigated nowadays in Discrete and Combinatorial Geometry are generated in the related area of Computational Geometry.
		Pach, P. Agarwal, Combinatorial Geometry, Wiley and Sons, 1995.
	www.dma.fi.upm.es /english/bienio99-01/geometriadiscreta/home.html (141 words)

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