By means of Parallel Coordinates planar "graphs" of multivariate relations are obtained. Certain properties of the relationship correspond to the geometrical properties of its graph. On the plane a point +----. line duality with several interesting properties is induced. A new duality between bounded and unbounded convex sets and hstars (a generalization of hyperbolas) and between Convex Unions and Intersections is found. This motivates some efficient Convexity algorithms and other results in Computational Geometry. There is also a suprising "cusp" *--~ "inflection point" duality. The narrative ends with a preview of the corresponding results in R N.

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Automation has arrived to Parallel Coordinates. A geometrically motivated classifier is presented and applied, with both training and testing stages, to 3 real datasets. Our results compared to those from 23 other classifiers have the least error. The algorithm is based on parallel coordinates and :has very low computational complexity in the number of variables and the size of the dataset -contrasted with the very high or unknown (often unstated) complexity of other classifiers, the low complexity enables the rule derivation to be done in near real-time hence making the classification adaptive to changing conditions, provides comprehensible and explicit rules -contrasted to neural networks which are "black boxes", does dimensionality selection -where the minimal set of original variables (not transformed new variables as in Principal Component Analysis) required to state the rule is found, orders these variables so as to optimize the clarity of separation between the designated set and its complement -this solves the pesky "ordering problem" in parallel coordinates.The algorithm is display independent, hence it can be applied to very large in size and number of variables datasets. Though it is instructive to present the results visually, the input size is no longer display-limited as for visual data mining.