CGGC Seminar: Visualizing R^N – Representing Surfaces by their Normal Vectors

Alfred Inselberg (Mathematics, Tel Aviv University)
Sunday, 20.6.2010, 11:00
Taub 401

With parallel coordinates the perceptual barrier imposed by our 3-dimensional habitation is breached enabling the visualization of multidimensional problems. By learning to recognize patterns a powerful knowledge discovery process evolved. It leads to a deeper geometrical insight: the recognition of M-dimensional objects recursively from their (M-1)-dimensional subsets. A hyperplane in N-dimensions is represented by (N -1) indexed points. Points representing lines have two indices, those representing planes three indices and so on. In turn, this yields powerful geometrical algorithms (e.g. for intersections, containment, proximities) and applications including classification. A smooth surface in 3-D is the envelope of its tangent planes each represented by 2 planar points. As a result it is represented by two planar regions, and a hypersurface in N-dimensions by (N-1) regions. This is equivalent to representing a surface by its normal vectors. Developable surfaces are represented by curves revealing the surface characteristics. Convex surfaces in any dimension are recognized by hyperbola-like regions. Non-orientable surfaces yield stunning patterns unlocking new geometrical insights. Non-convexities like folds, bumps, concavities are not hidden. The patterns persist in the presence of errors and that’s good news for applications opening the way for the exploration of massive datasets. Applications of parallel coordinates include collision avoidance and conflict resolution algorithms for air traffic control (3 USA patents), computer vision (USA patent), data mining (USA patent), decision support and process control.

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