![](bg8.png)
it follows that the hyper-surface σ can be mapped
(i.e. represented) by n − 1 indexed planar regions
¯σ
i
composed of these points. Restricted classes of
hyper-surfaces have been represented in this way and
it turns out that the ¯σ
i
reveal non-trivial properties
of the corresponding hyper-surface σ. It has already
been proved that for any dimension Quadrics (alge-
braic surfaces of degree 2) are mapped into planar re-
gions whose boundaries are conics (Izhakian, 2004).
This also includes non-convex surfaces like the “sad-
dle”. There are strong evidence supporting the con-
jecture that algebraic surfaces in general map into pla-
nar regions bounded by algebraic curves. This, be-
sides their significance on their own right, is one of
the reasons for studying the representation of alge-
braic curves. In (Inselberg, 2000) families of approx-
imate planes and flats are beautifully and usefully rep-
resented in -coords. Our results cast the foundations
not only for the representation of hyper-surfaces in
the class (S), but also their approximations in terms
of planar curved regions.
REFERENCES
B. Dimsdale, “Conic transformations and projectivities”,
IBM Los Angeles Scientific Center, 1984, Rep. G320-
2753.
A. Inselberg and T. Matskewich, “Approximated planes
in parallel coordinates”, Vanderbilt University Press,
Paul Sabloniere Pierre-Jean Laurent and Larry L. Shu-
maker (eds.), Eds., 2000, pp. 257–267.
Z. Izhakian, “An algorithm for computing a polynomial’s
dual curve in parallel coordinates”, M.sc thesis, De-
partment of Computer Science, University of Tel Aviv,
2001.
Z. Izhakian, “New Visualization of Surfaces in Parallel Co-
ordinates - Eliminating Ambiguity and Some Over-
Plotting”, Journal of WSCG - FULL Papers Vol.1-3,
No.12, ISSN 1213-6972, 2004, pp 183-191.
A. Inselberg, “The plane with parallel coordinates”, The
Visual Computer, vol. 1, no. 2, pp. 69–92, 1985.
A. Inselberg, “Don’t panic ... do it in parallel!”, Computa-
tional Statistics, vol. 14, pp. 53–77, 1999.
R. L. Burden and J. D. Faires, “Numerical analysis”, 4th
ed, PWS-Kent, Boston, MA, 1989.
D. Cox, J. Little, and D. O’Shea, “Ideals, Varieties, and
Algorithms”, Springer, New York, second ed. edition,
1997.
G. b. Folland, “Real analysis: modern techniques and their
applications”, Wiley, New York, second ed. 1999.
J. Harris, “Algebraic geometry”, A first course, Springer-
Verlag, New York, 1992.
W. Hodge and D. Pedoe, “Methods of algebraic geometry”,
Vol. II. Cambridge: Cambridge Univ. Press, 1952.
R. J. Walker, “Algebraic Curves”, Springer-Verlag, New
York, 1978.
G. Walter, “Numerical analysis : an introduction”,
Birhauser, Boston, 1997.
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