Alfred Inselberg

The plane with parallel coordinates

By means ofParallel Coordinates planar “graphs” of multivariate relations are obtained. Certain properties of the relationship correspond tothe geometrical properties of its graph. On the plane a point ←→ line duality with several interesting properties is induced. A new duality betweenbounded 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 inComputational Geometry. There is also a suprising “cusp” ←→ “inflection point” duality. The narrative ends with a preview of the corresponding results inRN.

Parallel coordinates: a tool for visualizing multi-dimensional geometry

A methodology for visualizing analytic and synthetic geometry in RN is presented. It is based on a system of parallel coordinates which induces a non-projective mapping between N-Dimensional and 2-Dimensional sets. Hypersurfaces are represented by their planar images which have some geometrical properties analogous to the properties of the hypersurface that they represent. A point ← → line duality when N = 2 generalizes to lines and hyperplanes enabling the representation of polyhedra in RN. The representation of a class of convex and non-convex hypersurfaces is discussed together with an algorithm for constructing and displaying any interior point. The display shows some local properties of the hypersurface and provides information on the points proximity to the boundary. Applications to Air Traffic Control, Robotics, Computer Vision, Computational Geometry, Statistics, Instrumentation and other areas are discussed.

Multidimensional lines I: representation

Based on a system of parallel coordinates, a methodology for visualizing multivariate relations (i.e., subsets of

Multidimensional lines II: proximity and applications

R^N

Noncommutative superpositions for nonlinear operators

) by mapping them uniquely into indexed subsets of

A mathematical model of the basilar membrane

R^2

An integral equation arising in a convective heat (mass) transfer problem through a boundary layer

was introduced earlier. Here, the foundations for the representation mapping in general are given, and the representation of lines in

Conflict detection and resolution between moving objects

R^N

System and method for visually querying a data set exhibited in a parallel coordinate system

by

N-Dimensional information display method for air traffic control

N - 1