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Geometry
Geometry is the branch of mathematics dealing with spatial relationships. From experience, or possibly intuitively, people characterize space by certain fundamental qualities, which are termed axioms in geometry. Such axioms are insusceptible of proof, but can be used in conjunction with mathematical definitions for points, straight lines, curves, surfaces, and solids to draw logical conclusions.
Because of its immediate practical applications, geometry was one of the first branches of mathematics to be developed. Likewise, it was the first field to be put on an axiomatic basis, by Euclid. The Greeks were interested in many questions about ruler-and-compass constructions. The next most significant development had to wait until a millennium later, and that was analytic geometry, in which coordinate systems are introduced and points are represented as ordered pairs or triples of numbers. This sort of representation has since then allowed us to construct new geometries other than the standard Euclidean version.
The central notion in geometry is that of congruence. In Euclidean geometry, two figures are said to be congruent if they are related by a series of reflections, rotations, and translationss.
Other geometries can be constructed by choosing a new underlying space to work with (Euclidean geometry uses Euclidean space, Rn) or by choosing a new group of transformations to work with (Euclidean geometry uses the inhomogeneous orthogonal transformations, E(n)). The latter point of view is called the Erlangen program. In general, the more congruences we have, the fewer invariants there are. As an example, in affine geometry any linear transformation is allowed, and so the first three figures are all congruent; distances and angles are no longer invariants, but linearity is.
A discrete form of geometry is treated under Pick's theorem.
See also List of geometry topics, Important publications in geometry.
External links
- Geometry. From Interactive Mathematics Miscellany and Puzzles
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HULL
Ken ClarkSon's convex hulls, Delaunay triangulations, alpha shapes calculator in C
http://cm.bell-labs.com/netlib/voronoi/hull.html
Magic Software
Dave Eberley's Code Page: Includes lot of Code that you might need while writing a Computational Geometric Software
http://www.magic-software.com/
Polygon Boolean operations on sets of polygons
GDSII viewer + Algorithms for 2d polygon boolean operations, C++ source code
http://www.xs4all.nl/~kholwerd/bool.html
Computational Geometry Code
Jeff Erickson's links to freely available implementations of geometric algorithms and software.
http://compgeom.cs.uiuc.edu/~jeffe/compgeom/code.html
Fast Industrial-Strength Triangulation
By Martin Held. Copes with polygons with holes and degeneracies/self-intersections; based on repeatedly clipping ears. Source code on request.
http://www.cosy.sbg.ac.at/~held/projects/triang/triang.html
QMG project
Mesh generation in 2D and 3D on Unix and NT, and related software by Steven Vavasis.
http://www.cs.cornell.edu/Info/People/vavasis/qmg-home.html
Delaunay Triangulation Code
J-D. Boissonnat et al.'s code for Delaunay meshing in 2 and 3 dimensions in C++.
http://www-sop.inria.fr/prisme/logiciel/del-tree.html
Fast Robust Predicates for Computational Geometry
C code for orientation and incircle tests using adaptive-precision floating-point arithmetic
http://www.cs.cmu.edu/~quake/robust.html
Stony Brook Algorithm Repository - Section on Computational Geometry
Summary of major CG problems and implementations of their solutions
http://www.cs.sunysb.edu/~algorith/major_section/1.6.shtml
JeoEdit
Two Java applets for editing polygons and point sets for input to computational geometry software.
http://cgm.cs.mcgill.ca/~godfried/jeoedit/
Gmsh
Gmsh is a three-dimensional finite element mesh generator, primarily Delaunay, with built-in pre- and post-processing facilities. Gmsh is freely available for Windows and most UNIX platforms.
http://www.geuz.org/gmsh/
Reviver Homepage
A Free Surface Reconstructor which turns unorganised point clouds to surfaces. Variety of Output Formats. (Surface Reconstruction Software)
http://www.ams.sunysb.edu/~piyush/reviver
GTS - The GNU Triangulated Surface Library
Open-source library for manipulating 3D triangular surface meshes. Delaunay triangulations, robust geometric predicates, robust boolean set operations on surfaces, surface refinement and coarsening, level-of-detail, k-d trees, volume and curvature, strips generation.
http://gts.sourceforge.net/
Edgebreaker 3D Compression for Triangle Meshes
Source code, explanations, file formats, and examples of the Edgebreaker compression and decompression techniques.
http://www.gvu.gatech.edu/~jarek/edgebreaker/eb/
Knotplot
A program for exploring topological knots and knot-like objects in a 3-D rendered environment offering stills, animated knots (including dynamic forces) and demos. Aimed at mathematician and artist alike.
http://www.cs.ubc.ca/nest/imager/contributions/scharein/KnotPlot.html
Surface Evolver
Free interactive program for modelling liquid surfaces shaped by various forces and constraints, e.g., minimal surfaces (bubbles). Applies gradient descent to minimize energy. Available for most platforms.
http://www.susqu.edu/facstaff/b/brakke/evolver/
Triangle
Jonathan Richard Shewchuk's Two-Dimensional Quality Mesh Generator and Delaunay Triangulator.
http://www.cs.cmu.edu/~quake/triangle.html
CM2 Mesh Tools
Professional C++ mesh generators. Includes product information and news releases.
http://www.computing-objects.com/en/meshtools.html
Knotenpunkt
Produces commercial software that creates a surface model out of a pointcloud or polymesh [Balingen, Germany].
http://www.knotenpunkt.com/home_E.htm
Smallest Enclosing Ball of Points
Fast and robust C++ code available under GNU license. Handles arbitrary dimensions, and has high numerical stability.
http://www.inf.ethz.ch/personal/gaertner/miniball.html
Polymake
A tool for the algorithmic treatment of polytopes and polyhedra.
http://www.math.tu-berlin.de/polymake/
Amenta's Directory of Computational Geometry Software
Lot of categories and links.
http://www.geom.uiuc.edu/software/cglist/
Polyhedral FAQ
Frequently Asked Questions in Polyhedral Computation.
http://www.ifor.math.ethz.ch/staff/fukuda/polyfaq/polyfaq.html
CGAL - Computational Geometry Algorithms Library
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http://www.cgal.org/
Surface Reconstruction Algorithms Page
Links to Surface Reconstruction software, papers, people.
http://www.compgeom.com/www.sites.html
atlas - differential geometry Maple package
Calculations in terms of manifolds, mappings, embeddings, submersions, p-forms, tensor fields etc. All calculations are as coordinate free as possible. Some calculations with non-numerical dimension are available. Modern differential geometry notations.
http://www.graphtree.com/Maple/atlas/index.htm
FastGEO Computational Geometry Library
FastGEO, A computational geometry library written in the object pascal language.
http://www.partow.net/projects/fastgeo/index.html
Mg R-tree Library
A simple C++ R-tree implementation with Segment Incidences Detector as an application.
http://www.volny.cz/r-tree/
Qhull - convex hulls, Delaunay triangulations, Voronoi diagrams, and halfspace intersections
Qhull computes convex hulls, Delaunay triangulations, Voronoi diagrams, half-space intersections about a point, furthest-site Delaunay triangulations, and furthest-site Voronoi diagrams. It runs in 2-d, 3-d, 4-d, and higher dimensions.
http://www.qhull.org/
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