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Representation of algorithms in forms of geometry is known as Computational Geometry. Computational geometry is used in the robotics field, computer aided engineering, geographic information simulation, integrated circuit design etc. Computational geometry is of two types:
- Combinatorial computational geometry – Deals with the geometric objects
- Numerical computational geometry - Represents the real world objects, known as geometric modeling and Computer aided geometric design
It contains the open problem and provides the short report on the implementations of computational geometry tools.
Few topics :
- Gift-wrapping algorithm
- Computing the convex hull
- Bentley ottmann's plane sweep
- Doubly-connected edge list
- Map overlay and application
- Douglas-peucker algorithm
- Polygon triangulation and art gallery problems
- Combinatorial bounds and properties
- Triangulating a monotone polygon
- Linear programming in 2d and 3d
- Divide-and-conquer algorithm
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- Surface modeling, B-splines, non-uniform rational b-splines, Physically based deformable surfaces, Sweeps and generalized cylinders, Offsets, blending and filleting surfaces, Non-linear solvers and intersection problems, Solid modeling, Constructive solid geometry, Boundary representation, Non-manifold and mixed-dimension boundary, Octrees, Robustness of geometric computations, Interval methods, Finite and boundary element discretization, Scientific visualization, Variational geometry, Tolerances. Inspection methods, Feature representation and recognition, Shape interrogation for design, analysis, and manufacturing, Involves analytical and programming
- Planar convex hull as a prototypical problem, Goals: Correctness, space and time efficiency, Basic planar problems, Convex hull, Line segment intersection, Map overlay, Applications to clipping and polygon intersection, Triangulating a simple polygon, art gallery problem: How many security cameras are needed?, Finding things – geometric query problems, Range searching, Organize data geometrically for fast queries, Preprocessing / query time trade-off, Point location, Trapezoidal decompositions, DAG search structures, Special planar decompositions
- Voronoi decomposition of a point set, Definition and properties, Fortune’s sweep algorithm, Generalized Voronoi decompositions (segments, weights), Medial axis and shape recognition applications, Delaunay triangulations, Duality, Angle optimality, Convex hull in R, Complexity, Incremental algorithm and analysis, quickhull algorithm, Computing 2D Voronoi with 3D convex hull, Binary space partition, Background: Application to rendering 3D graphics, Construction of a BSP tree, Pathological configurations and size bounds, Motion planning(a) Background: Application to robotics, Configuration spaces, Translating polygonal robot case, Minkowski sums, Allowing rotation, Finding shortest paths, Computing visibility graphs, computer graphics and animation, computer vision, computer-aided design and manufacturing, geographic information systems, pattern recognition, wireless communications, robotics, protein folding, urban planning, graph drawing, statistical analysis, to name just a few, combinatorial and discrete geometry, Geometric searching, Convex hulls, Proximity computations, Intersections, Graph drawing, Computer graphics, Computer vision, Software engineering, Databases, robotics, Geographical information systems.
Help for complex topics like :
- Data structures, algorithms, analysis techniques for computational problems, Line segment intersection, polygon triangulation, 2D linear programming, range queries, point location, arrangements , duality, Voronoi diagrams , Delaunay triangulation, convex hulls, robot motion planning, visibility graphs, Polygon triangulations and partitions, Convex hulls, Delaunay and Voronoi diagrams, Arrangements, Spatial queries, KD-trees, BSP-trees, Quadtrees, Convex Hulls, Triangulation and the art gallery problem, Binary Space Partitions, Voronoi Diagrams and Delaunay Triangulations, Quadtrees, Kd-trees, Distance computation in 2D and 3D,Motion planning in 2D, Visibility Graphs, Line segment intersection, Triangulation, Linear programming
- Range search, Point location, Voronoi diagram, Arrangement and duality, Visibility graph, Well separated pair decomposition, VC-dimension, -approximation, and nets, Geometry foundations: motivation, primitives, transformation, Polygons and Art Gallery Theorem, Convex hull: convex hull in 2D and 3D, and applications, Triangulations and Voronoi diagrams: Delaunay triangulation and special cases; graphs, 2D, 3D and weighted constructions, duality, Curves: medial axis, reconstruction, Tetrahedron meshes, Polyhedra, Surfaces: reconstruction, surface simplification, Algorithmic Background
- Geometric Preliminaries, Models of Computation, Geometric Searching, Point-Location Problems, Range-Searching Problems, Convex Hulls, Preliminaries, Problem Statement and Lower Bounds, Convex Hull Algorithms in the Plane, Graham's Scan, Jarvis's March, QUICKHULL techniques, Dynamic Convex Hull, Convex Hull in 3D, Proximity Problem, A Collection of Problems, A Computational Prototype: Element Uniqueness, Lower Bounds,Closest-Pair Problem: A Divide-and-Conquer Approach,Voronoi Diagram, Proximity Problems Solved by the Voronoi Diagram, Triangulation, Planar Triangulations, Greedy Triangulations, Partitioning a Polygon into Monotone Pieces, Triangulating a Monotone Polygon, Delaunay Triangulation, Intersections, Application Areas, Planar Applications: Intersection of Convex Polygons, Star-shaped Polygons; Intersection of Line Segments, 3D Applications: Intersection of 3D Convex Polyhedra; Intersection of Half-spaces.
- Convex hulls,Voronoi diagrams,Delaunay triangulations,Euclidean Minimum Spanning Trees ,Geometric algorithms for Linear programming,Range Searching with database applications,Point location,Combinatorial geometry of arrangements,Shortest paths with applications in Robotics,BSP-trees and applications in graphics
Few Topics are:
- algorithms and data structures
- convex hulls
- Voronoi diagrams
- Delauney triangulation
- Euclidean spanning trees
- point location, and range searching
- lower bounds and discrepancy theory
- Optimization in geometry
- fixed dimensional linear programming and shortest paths
- Graphic data structures, BSP trees