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Paper

ALGEBRAIC CURVES IN PARALLEL COORDINATES Avoiding the “Over-Plotting” ProblemTopics: Fundamental Methods and Algorithms; Model Validation; Modeling and Algorithms; Sketch-Based Modeling

Keyword(s):Scientific Visualization, Multi-dimensional Geometric Modeling and Algorithms, Parallel Coordinates.

Related
Ontology
Subjects/Areas/Topics:Computer Vision, Visualization and Computer Graphics
;
Fundamental Methods and Algorithms
;
Geometry and Modeling
;
Model Validation
;
Modeling and Algorithms
;
Sketch-Based Modeling

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.
(More)

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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Izhakian Z. and (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

@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}, }

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