TY - GEN

T1 - Algebraic curves in Parallel Coordinates

T2 - GRAPP 2006 - 1st International Conference on Computer Graphics Theory and Applications

AU - Izhakian, Zur

PY - 2006

Y1 - 2006

N2 - Until 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 hyper-surfaces is the higher degree of the image's boundary which is also an algebraic curve in ||-coords.

AB - Until 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 hyper-surfaces is the higher degree of the image's boundary which is also an algebraic curve in ||-coords.

KW - Multi-dimensional geometric modeling and algorithms

KW - Parallel Coordinates

KW - Scientific visualization

UR - http://www.scopus.com/inward/record.url?scp=74949105827&partnerID=8YFLogxK

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AN - SCOPUS:74949105827

SN - 9728865392

SN - 9789728865399

T3 - GRAPP 2006 - Proceedings of the 1st International Conference on Computer Graphics Theory and Applications

SP - 150

EP - 157

BT - GRAPP 2006 - Proceedings of the 1st International Conference on Computer Graphics Theory and Applications

Y2 - 25 February 2006 through 28 February 2006

ER -