2 research outputs found
Algebraic Curves in Parallel Coordinates - Avoiding the "Over-Plotting" Problem
ntil 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 is an algebraic of
degree at most 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
Algebraic Curves in Parallel Coordinates – Avoiding the “Over-Plotting ” Problem
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 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