Graphics Reference
In-Depth Information
andadapted intothemoregeneralproblem-solving process.Weforma mental image
oftheproblemwearetryingtosolve,andattimeswesaythatwesee whenwe mean
that we understand.
My interest in visualization was also sparked and nourished while learning ge-
ometry. Later,while studying multidimensional geometry I became frustratedbythe
absence of visualization. I wondered why geometry was being tackled without using
(the fun and benefits of) pictures? Was there a generic way of creating accurate pic-
tures of multidimensional problems, like Descartes' epoch-making coordinate sys-
tem? What is “sacred” about orthogonal axes, which quickly “use up” the plane asso-
ciated with them? Ater all, parallelism rather than orthogonality is the fundamental
concept in geometry, and these are
not
equivalent terms, for orthogonality requires
a prior concept of “angle.” I played with the idea of a multidimensional coordinate
system based on parallel coordinates, in which, in principle, lots of axes could be
placed and viewed. Encouraged in
by Professors S.S. Cairns and D. Bourgin,
both topologists at the University of Illinois whereI was studying, I derived the basic
point
N-dimensional line correspon-
dence,realizing inthe processthat projective rather than Euclidean space isinvolved.
In
, while teaching linear algebra to a large class of unruly engineers, I was
challenged to show spaces with more than three dimensions. I recalled parallel co-
ordinates (abbreviated
line duality followed by the N
−
points
-coords), and the question of how multidimensional lines,
planes and other objects “look” was raised. his triggered the systematic develop-
ment of
-coords, and a comprehensive report Inselberg (
) documented the ini-
tial results. Noted on the first page is the superficial resemblance to nomography,
where the term “parallel coordinates” was used (Brodetsky,
). In nomography,
whichdeclinedwiththeadventofcomputers,therearegraphicalcomputationaltech-
niques involving “functional scales” (Otto,
, p.
), oten placed in parallel, for
problems with usually two or three variables. Until very recently, I was not aware
(I am indebted to M. Friendly forpointing this out in (Friendly,
))of d'Ocagne's
marvellous monograph (d'Ocagne,
)on parallel coordinates fortwo dimensions,
whereapoint
line-curve
correspondence, was applied to interesting numerical problems. D'Ocagne pursued
computational applications of nomography (d'Ocagne,
) rather than develop-
ing a multidimensional system of parallel coordinates - a systematic body of knowl-
edge consisting of theorems on the representation of multidimensional objects, their
transformations and geometrical construction algorithms (i.e., for intersections, in-
terior points, etc.), which is where we came in.
his is a good time to recoup and trace the development of
linedualitywasstudiedand,togetherwithapoint-curve
-coords. In
, it
was first presented at an international conference (Inselberg,
), and prior to that
at seminars like that at the University of Maryland, where I was fortunate to meet
Ben Shneiderman, whose encouragement and visualization wisdom I have benefited
from ever since. A century ater the publication of d'Ocagne's monograph, Inselberg
(
) (coincidentally) appeared. his period of research - performed in collabora-
tion with my esteemed colleagues the late B.Dimsdale (a long-time associate of John
von Neumann), A. Hurwitz, who was invaluable in every aspect leading to three US
patents (Collision Avoidance Algorithms for Air-Tra
c Control), Rivero and Insel-
