Alfred Inselberg (Mathematics, Tel Aviv University)
With parallel coordinates the perceptual barrier imposed by our
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
M-dimensional objects recursively from their (M-1)-dimensional subsets. A
hyperplane in N-dimensions is represented by (N -1) indexed 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
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
resolution algorithms for air traffic control (3 USA patents), computer
vision (USA patent), data mining (USA patent), decision support and process