Description of Surfaces in Parallel Coordinates by Linked Planar Regions.
ABSTRACT An overview of the methodology covers the representation (i.e. visualization) of multidimensional lines, planes, flats, hyperplanes,
and curves. Starting with the visualization of hypercubes of arbitrary dimension the representation of smooth surfaces is
developed in terms of linked planar regions. The representation of developable, ruled, non-orientable, convex and non-convex surfaces in ℝ3 with generalizations to ℝ
are presented enabling efficient visual detection of surface properties. The parallel coordinates methodology has been applied
to collision avoidance algorithms for air traffic control (3 USA patents), computer vision (1 USA patent), data mining (1
USA patent), optimization and elsewhere.
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ABSTRACT: We prove a Plücker formula,for a projective variety X with arbitrary singularities, which expresses the class of X, the degree of the dual variety, in terms of Euler characteristics of X and of two linear sections of X. Moreover, we show that there is no formula whatsoever expressing this degree as a difference of two terms, a deformation invariant and a correction for singularities.Communications in Algebra. 01/1997; 25(9):2897-2901.