2004/03/04 by Zur Izhakian
Computer Science · #cs.OH
published as Journal of WSCG, 2004, volume 1-3, no 12, pp 183-191, ISSN 1213-6972 · 13 pages 8 figures
arxiv created 2004/03/04 · arxiv updated 2009/12/01
\calA point P ∈ \Realn is represented in Parallel Coordinates by a polygonal line P (see \citeInsel99a for a recent survey). Earlier \citeinselberg85plane, a surface σ 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 σ as the envelope of it's \em tangent planes and in turn, representing each of these planes by n-1 points \citeInsel99a. This, with some future extension, can yield a new and unambiguous representation, σ, 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. \citehung92smooth) and \em classification criteria. It is further shown that the image (i.e. representation) of an algebraic surface of degree 2 in \Realn 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.