"Computational geometry is a branch of computer science devoted to the study of algorithms which can be stated in terms of geometry."
It examines algorithms and data structures for efficiently solving geometric problems, such as finding intersections, computing distances, and constructing shapes.
Cartesian coordinates and Euclidean geometry: This forms the foundation of computational geometry as most problems are formulated using geometric concepts and coordinates.
Polygon Triangulation: A fundamental algorithm in computational geometry, whose application includes convex hull, partitions, triangulations, Voronoi diagrams, and Delaunay triangulations.
Convex Hull: A computational geometry problem where an algorithm tries to find the smallest convex region that completely encloses a set of points.
Voronoi Diagrams: A Voronoi diagram divides space (usually 2D or 3D) into regions based on specific criteria, producing a partitioned graph.
Delaunay Triangulations: A Delaunay triangulation is a triangulation of space that satisfies certain conditions, such as ensuring that no vertex falls inside a circumcircle of any triangle.
Geometric Transformations: Used to transform geometric shapes, points, and lines.
Computational Complexity: The study of the amount of time, memory, or other resources necessary to solve a specific problem algorithmically.
Arrangements of Curves and Surfaces: The arrangement of a collection of curves and surfaces in a plane, 3D space, or higher-dimensional space.
Linear Programming and Convex Optimization: Linear programming is a specific method for solving optimization problems, particularly those that take place in on a plane or surface.
Topological Properties of Data and Algorithm Design: The topological properties of data refer to the properties of data that are independent of the details of the coordinate system being used, which can be important for algorithm design.
Convex Hull: Calculates the smallest convex polygon that contains all points in a given set.
Voronoi Diagrams: Breaks space into a set of regions based on the proximity to a set of points.
Delaunay Triangulation: Connects a set of points with non-overlapping triangles, where no other points exist inside the triangle.
Arrangements and Duality: Divides the plane into disjoint regions, each with a distinct property.
Motion Planning: Determines how to maneuver a robot in order to avoid obstacles and reach a target location.
Geometric Networks: Studies the connectivity properties of geometric objects, such as line segments or points.
Geometric Data Structures: Examines the organization and storage of geometric data, such as graphs and polygons.
Mesh Generation: Constructs a network of triangles to approximate the shape of a 3D object.
Visibility Graphs: Determines the visible areas between two or more points in space.
Computational Topology: Examines the shape and properties of objects and their relationship to each other.
Randomized Algorithms: Uses randomness to improve the speed and efficiency of geometric algorithms.
Algebraic Geometry: Applies algebraic techniques for solving geometric problems.
Surface and Curve Reconstruction: Generates a smooth approximation of a 3D shape from a set of data points.
Shape Analysis: Analyzes the properties of shapes, such as curvature or symmetry.
Geometric Optimization: Finds the best geometric shape or arrangement for a particular application or problem.
"While modern computational geometry is a recent development, it is one of the oldest fields of computing with a history stretching back to antiquity."
"Computational complexity is central to computational geometry, with great practical significance if algorithms are used on very large datasets containing tens or hundreds of millions of points."
"For such sets, the difference between O(n2) and O(n log n) may be the difference between days and seconds of computation."
"The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (CAD/CAM)."
"Some purely geometrical problems arise out of the study of computational geometric algorithms, and such problems are also considered to be part of computational geometry."
"Other important applications of computational geometry include robotics, geographic information systems (GIS), integrated circuit design, computer-aided engineering (CAE), and computer vision."
"The main branches of computational geometry are Combinatorial computational geometry (algorithmic geometry) and Numerical computational geometry (machine geometry, computer-aided geometric design, and geometric modeling)."
"A groundlaying book in the subject by Preparata and Shamos dates the first use of the term 'computational geometry' in this sense by 1975."
"This branch may be seen as a further development of descriptive geometry and is often considered a branch of computer graphics or CAD."
"The term 'computational geometry' in this meaning has been in use since 1971."
"Although most algorithms of computational geometry have been developed (and are being developed) for electronic computers, some algorithms were developed for unconventional computers (e.g., optical computers)."
"A groundlaying book in the subject by Preparata and Shamos dates the first use of the term 'computational geometry' in this sense by 1975."
"This branch may be seen as a further development of descriptive geometry and is often considered a branch of computer graphics or CAD."
"The main impetus for the development of computational geometry as a discipline was progress in computer graphics and computer-aided design and manufacturing (CAD/CAM)."
"Other important applications of computational geometry include robotics (motion planning and visibility problems)."
"Other important applications of computational geometry include geographic information systems (GIS) (geometrical location and search, route planning)."
"Other important applications of computational geometry include computer-aided engineering (CAE) (mesh generation)."
"Other important applications of computational geometry include computer vision (3D reconstruction)."
"For such sets, the difference between O(n2) and O(n log n) may be the difference between days and seconds of computation."