| Computational Geometry |
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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. Other important applications of computational geometry include Robotics (motion planning and visibility problems), Geographic Information System s (GIS) (geometrical location and search, route planning), Integrated Circuit design (IC geometry design and verification), computer-aided engineering (CAE) (programming of numerically controlled (NC) machines). The two main branches of computational geometry are:
The latter is often considered a branch of Computer Graphics and/or CAD , and the former is called simply ''computational geometry''. COMBINATORIAL COMPUTATIONAL GEOMETRY The primary goal of research in combinatorial computational geometry is to develop efficient s, Polyhedra , etc. Some of these problems seem so simple that they were not regarded as problems at all until the advent of Computer s. Consider, for example, the ''Closest pair problem'':
One could compute the distances between all the pairs of points, of which there are ''N''(''N'' − 1)/2, then pick the pair with the smallest distance. This Brute-force algorithm has a ''time complexity'' of '''O(''N''2)'''; i.e., its execution time is proportional to the square of the number of points. One milestone in computational geometry was the formulation of an algorithm with the smaller time complexity '''O(''N'' log ''N'')'''. For modern GIS, computer graphics, and integrated-circuit design systems routinely handling tens and hundreds of million points, the difference between ''N''2 and '''''N'' log ''N''''' is the difference between days and seconds of computation. Hence the emphasis on Computational Complexity in computational geometry. Problems ''main article: List Of Combinatorial Computational Geometry Topics .'' Core algorithms and problems Problems from this list have wide applications in areas processing of geometric information is used.
NUMERICAL GEOMETRY This branch is also known as Geometric Modelling , '''computer-aided geometric design''' (CAGD), and may be often found under the keyword '''curves and surfaces'''. Core problems are curve and surface modelling and representation. The most important instruments here are Parametric Curve s and Parametric Surface s, such as Bezier Curve s, Spline curves and surfaces. An important non-parametric approach is the Level Set Method . First (and still most important) application areas are shipbuilding, aircraft, and automotive industries. However because of modern ubiquity and power of computers even perfume bottles and shampoo dispensers are designed using techniques unheard of by shipbuilders of 1960s. SEE ALSO
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