  $\cal{A}$ point $P \in \Real^n$ is represented in Parallel Coordinates by a
polygonal line $\bar{P}$ (see \cite{Insel99a} for a recent survey). Earlier
\cite{inselberg85plane}, a surface $\sigma$ was represented as the {\em
envelope} of the polygonal lines representing it's points. This is ambiguous in
the sense that {\em different} surfaces can provide the {\em same} envelopes.
Here the ambiguity is eliminated by considering the surface $\sigma$ as the
envelope of it's {\em tangent planes} and in turn, representing each of these
planes by $n$-1 points \cite{Insel99a}. This, with some future extension, can
yield a new and unambiguous representation, $\bar{\sigma}$, of the surface
consisting of $n$-1 planar regions whose properties correspond lead to the {\em
recognition} of the surfaces' properties i.e. developable, ruled etc.
\cite{hung92smooth}) and {\em classification} criteria.
  It is further shown that the image (i.e. representation) of an algebraic
surface of degree 2 in $\Real^n$ is a region whose boundary is also an
algebraic curve of degree 2. This includes some {\em non-convex} surfaces which
with the previous ambiguous representation could not be treated. An efficient
construction algorithm for the representation of the quadratic surfaces (given
either by {\em explicit} or {\em implicit} equation) is provided. The results
obtained are suitable for applications, to be presented in a future paper, and
in particular for the approximation of complex surfaces based on their {\em
planar} images. An additional benefit is the elimination of the
``over-plotting'' problem i.e. the ``bunching'' of polygonal lines which often
obscure part of the parallel-coordinate display.

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