Visualization provides insight through images , and can be considered as a collection of application specific mappings: problem domain → visual range. For the visualization of multivariate problems a multidimensional system of Parallel Coordinates is studied which provides a one-to-one mapping between subsets of N-space and subsets of 2-space. The result is a systematic and rigorous way of doing and seeing analytic and synthetic N-dimensional geometry. Lines in N-space are represented by N-1 indexed points. In fact all p-flats (planes of dimension p in N-space) are represented by indexed points where the number of indices depends on p and N. The representations are generalized to enable the visualization of polytopes and certain kinds of hypersurfaces as well as recognition of convexity. Several algorithms for constructing and displaying intersections, proximities and points interior/exterior/or on a hypersurface have been obtained. The methodology has been applied to visual data mining, process control, medicine, finance, retailing, collision avoidance algorithms for air traffic control, optimization and others.
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