Parallel Coordinate Representations of Smooth Hypersurfaces Alfred Inselberg#
Chao-Kuei Hung#
[email protected]
[email protected]
IBM Scienti c Center of Computer Science 2525 Colorado Avenue University of Southern California Santa Monica, CA 90404 Los Angeles, CA 90089{0782 December 11, 1992
# Department
1 Introduction Visualizing (or imagining) problems which may not be intrinsically geometric has been an invaluable source of ingenuity in human discoveries and inventions. As the dimension of the (abstracted) problem goes from 2 to 3, however, visualization becomes dicult; and our intuition staggers for dimensions beyond. Our goal in the parallel coordinate representation is to nd a mapping from 2RN to 2R2 such that geometric features of an object in RN become geometric features of its image in R under this mapping. At rst sight there seems no way for this mapping to be unambiguous for there seems to be much more objects in RN than there are in R . A second thought reveals that, however, it can be injective (1-to-1) in principle since the cardinality of the domain is the same as that of the range, both being @ . Rather than trying to deal with arbitrary subsets of RN we rst restrict ourselves to subsets that most real-world problems deal with. Inselberg [6], Dimsdale [7], and Eickemeyer [2] gave several results for general linear subsets which were applied to many practical situations such as air trac control, medicine, and others [9] [3]. In this paper we further this representation paradigm by applying those results to smooth surfaces. We rst present an encouraging result which enables us to unambiguously represent developable surfaces in R3. Then a surprisingly straightforward reconstruction procedure (given the parallel coordinate representation) is presented and proved. Cones and cylinders { the most \popular" developable surfaces arising in engineering and manufacturing { are looked at more closely. The e ect of their orientations on their representations is studied. Partial results for a more general class of smooth surfaces { the ruled surfaces { are also given. We nally conclude with possible generalizations and implications of this work. For readers not familiar with parallel coordinates the previous relevant results are summarized in the appendix, which also gives the notation used throughout this paper and is di erent from Eickemeyer's. 2
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Most of our results require that the functions and/or parameterizations be smooth (C unless otherwise speci ed).
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2 2- at Representation for Developable Surfaces Our rst goal is to represent smooth surfaces in the parallel coordinate plane. With Eickemeyer's representations for ats at hand, what could be more natural than trying to identify a smooth surface with it's tangent planes (or tangent 2- ats in our terminology) ? We begin by recalling that a smooth surface in R is generally the envelope of a 2-parameter family of 2- ats { its tangent 2- ats [4]. Thus we would expect to get a 2-parameter family of points on the parallel coordinate plane in general were we to plot, say, for every plane tangent to a given surface . Of particular interest is the case where has only a 1-parameter family of tangent 2- ats, namely that it is a developable surface . Lemma 2.1 (One-Parameter Family of Planes) Let (t) : n(t) R = (t) be a one parameter family of 2- ats in R3 such that ni (t) for i = 1; 2; 3 and (t) are smooth. Let U be some neighborhood of the point such that 3
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