Algebraic Curves in Parallel Coordinates - Avoiding the "Over-Plotting" Problem

TL;DR

This paper introduces a transformation for representing algebraic curves in parallel coordinates that avoids over-plotting by directly producing a point-curve representation, extending previous work to all planar algebraic curves.

Contribution

It extends the dual representation method to all planar algebraic curves and provides an algebraic geometry-based algorithm for constructing the dual image.

Findings

Dual of an algebraic curve of degree n has degree at most n(n-1).

The method effectively reduces over-plotting in parallel coordinates.

Algorithm uses resultants and homogeneous polynomials for curve dualization.

Abstract

${\cal U}$ntil now the representation (i.e. plotting) of curve in Parallel Coordinates is constructed from the point $\leftrightarrow$ line duality. The result is a ``line-curve'' which is seen as the envelope of it's tangents. Usually this gives an unclear image and is at the heart of the ``over-plotting'' problem; a barrier in the effective use of Parallel Coordinates. This problem is overcome by a transformation which provides directly the ``point-curve'' representation of a curve. Earlier this was applied to conics and their generalizations. Here the representation, also called dual, is extended to all planar algebraic curves. Specifically, it is shown that the dual of an algebraic curve of degree $n$ is an algebraic of degree at most $n(n - 1)$ in the absence of singular points. The result that conics map into conics follows as an easy special case. An algorithm, based on algebraic geometry using resultants and homogeneous polynomials, is obtained which constructs the dual image of the curve. This approach has potential generalizations to multi-dimensional algebraic surfaces and their approximation. The ``trade-off'' price then for obtaining {\em planar} representation of multidimensional algebraic curves and hyper-surfaces is the higher degree of the image's boundary which is also an algebraic curve in $\|$-coords.