  ${\cal U}$ntil now the representation (i.e. plotting) of curve in Parallel
Coordinates is constructed from the point $\leftrightarrow$ 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.
