2004/03/04 by Zur Izhakian, Izhakian, Zur
Computer Science · #Constraint Satisfaction and Optimization #Data Management and Algorithms #Data Visualization and Analytics #F.2.1. #FOS: Computer and information sciences #I.1.1 #Other Computer Science (cs.OH) #cs.OH
paper · pdf · doi:10.48550/arxiv.cs/0403005
arxiv created 2004/03/04 · openalex publication_date 2004/03/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
\cal Until now the representation (i.e. plotting) 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 \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.