New visualization of surfaces in parallel coordinates: eliminating ambiguity and some “over-plotting”

Abstract

A point P 2 Rn is represented in Parallel Coordinates by a polygonal line ¯ P (see [Ins99] for a recent survey). Earlier [Ins85], a surface ¾ was represented as the envelope of the polygonal lines representing it’s points. This is ambiguous in the sense that different surfaces can provide the same envelopes. Here the ambiguity is eliminated by considering the surface ¾ as the envelope of it’s tangent planes and in turn, representing each of these planes by n-1 points [Ins99]. This, with some future extension, can yield a new and unambiguous representation, ¯¾, of the surface consisting of n-1 planar regions whose properties correspond lead to the recognition of the surfaces’ properties i.e. developable, ruled etc. [Hun92]) and classification criteria. It is further shown that the image (i.e. representation) of an algebraic surface of degree 2 in Rn is a region whose boundary is also an algebraic curve of degree 2. This includes some 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 explicit or 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 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.