Zur Izhakian

Tel Aviv University

Department of Computer Science

Ramat Aviv, 69978, Tel Aviv,

Israel

zzur@post.tau.ac.il: |
http:// |

**Abstract**

A point *P* in *R^n* is represented in Parallel Coordinates by a polygonal line *P* (see [Ins99] for a recent survey). Earlier [Ins85], a surface *S* 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 *S* as the envelope of it's * tangent planes * and in turn, representing each of these planes by *n-1* points [Ins99]. This yields a new and unambiguous representation, *S*, 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 *R^n* 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.