By learning to untangle patterns from the displays; a powerful knowledge discovery process has evolved. It is illustrated on real datasets together with guidelines for exploration and good query design. Realizing that this approach is intrinsically limited leads to a deeper geometrical insight; the recognition of M-dimensional objects recursively from their (M – 1)-dimensional subsets. It emerges that any linear N-dimensionsal relation is represented by (N – 1) indexed points. The indexing in ||-coords is not well understood and will be demystified. Indexing enables the concentration of relational information into patterns and paves the way for coping with large datasets. For example in 3-D; two points with two indices represent a line and two points with three indices represent a plane. There result powerful geometrical algorithms (intersections; containment; proximities) and applications including classification.
A smooth surface is the envelope of its tangent planes. This is equivalent to representing the surface by its normal vectors; rather than projections as in standard surface descriptions. Developable surfaces are represented by curves revealing the surfaces’ characteristics. Convex surfaces in any dimension are recognized by the hyperbolalike (i.e. having two asymptotes) regions from just one orientation. Non-orientable surfaces (i.e. like the Möbius strip) yield stunning patterns unlocking new geometrical insights. Non-convexities like folds; bumps; concavities and more are no longer hidden and are detected from just one orientation. Evidently this representation is preferable for some applications even in 3-D. The patterns persist in the presence of errors deforming in ways revealing the type and magnitude of the errors and that’s good news for the applications. We stand on the threshold of cracking the gridlock of multidimensional visualization.
The parallel coordinates methodology is used in collision avoidance and conflict resolution algorithms for air traffic control (3 USA patents); computer vision (USA patent); data mining (USA patent) for data exploration and classification; multiobjective optimization; decision support; and process control.