Parallel coordinates: a tool for visualizing multi-dimensional geometry

TLDR

Hypersurfaces are represented by planar images whose geometric properties mirror those of the hypersurfaces they depict. The paper presents a methodology for visualizing analytic and synthetic geometry in RN. The method uses parallel coordinates to map N‑dimensional geometry to 2‑D, employing point‑line duality and an algorithm that constructs and displays interior points of convex and non‑convex hypersurfaces. The display reveals local properties and boundary proximity of hypersurfaces, and the technique is applicable to air‑traffic control, robotics, computer vision, computational geometry, statistics, and instrumentation.

Abstract

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 point's proximity to the boundary. Applications to Air Traffic Control, Robotics, Computer Vision, Computational Geometry, Statistics, Instrumentation and other areas are discussed.