# ALGEBRAIC CURVES IN PARALLEL COORDINATES Avoiding the “Over-Plotting” Problem

### Zur Izhakian

#### Abstract

U ntil now the representation (i.e. modeling) of curve in Parallel Coordinates is constructed from the point ↔ 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 planar representation of multidimensional algebraic curves and hypersurfaces is the higher degree of the image’s boundary which is also an algebraic curve in -coords.

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#### in Harvard Style

Izhakian Z. (2006). ALGEBRAIC CURVES IN PARALLEL COORDINATES Avoiding the “Over-Plotting” Problem . In Proceedings of the First International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP, ISBN 972-8865-39-2, pages 150-157. DOI: 10.5220/0001356601500157

#### in Bibtex Style

@conference{grapp06,
author={Zur Izhakian},
title={ALGEBRAIC CURVES IN PARALLEL COORDINATES Avoiding the “Over-Plotting” Problem},
booktitle={Proceedings of the First International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,},
year={2006},
pages={150-157},
publisher={SciTePress},
organization={INSTICC},
doi={10.5220/0001356601500157},
isbn={972-8865-39-2},
}

#### in EndNote Style

TY - CONF
JO - Proceedings of the First International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,
TI - ALGEBRAIC CURVES IN PARALLEL COORDINATES Avoiding the “Over-Plotting” Problem
SN - 972-8865-39-2
AU - Izhakian Z.
PY - 2006
SP - 150
EP - 157
DO - 10.5220/0001356601500157