# ALGEBRAIC CURVES IN PARALLEL COORDINATES Avoiding the “Over-Plotting” Problem

### Zur Izhakian

#### Abstract

U ntil now the representation (i.e. modeling) of curve in Parallel Coordinates is constructed from the point ↔ line duality. The result is a “line-curve” which is seen as the envelope of it’s tangents. Usually this gives an unclear image and is at the heart of the “over-plotting” problem; a barrier in the effective use of Parallel Coordinates. This problem is overcome by a transformation which provides directly the “point-curve” representation of a curve. Earlier this was applied to conics and their generalizations. Here the representation, also called dual, is extended to all planar algebraic curves. Specifically, it is shown that the dual of an algebraic curve of degree n is an algebraic of degree at most n(n − 1) in the absence of singular points. The result that conics map into conics follows as an easy special case. An algorithm, based on algebraic geometry using resultants and homogeneous polynomials, is obtained which constructs the dual image of the curve. This approach has potential generalizations to multi-dimensional algebraic surfaces and their approximation. The “trade-off” price then for obtaining planar representation of multidimensional algebraic curves and hypersurfaces is the higher degree of the image’s boundary which is also an algebraic curve in -coords.

#### 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. Shumaker (eds.), Eds., 2000, pp. 257-267.
- Z. Izhakian, “An algorithm for computing a polynomial's dual curve in parallel coordinates”, M.sc thesis, Department of Computer Science, University of Tel Aviv, 2001.
- Z. Izhakian, “New Visualization of Surfaces in Parallel Coordinates - Eliminating Ambiguity and Some OverPlotting”, 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!”, Computational 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, SpringerVerlag, 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.
- Walter, “Numerical analysis : an introduction”, Birhauser, Boston, 1997.

#### Paper Citation

#### in Harvard Style

Izhakian Z. (2006). **ALGEBRAIC CURVES IN PARALLEL COORDINATES Avoiding the “Over-Plotting” Problem** . In *Proceedings of the First International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,* ISBN 972-8865-39-2, pages 150-157. DOI: 10.5220/0001356601500157

#### in Bibtex Style

@conference{grapp06,

author={Zur Izhakian},

title={ALGEBRAIC CURVES IN PARALLEL COORDINATES Avoiding the “Over-Plotting” Problem},

booktitle={Proceedings of the First International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,},

year={2006},

pages={150-157},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0001356601500157},

isbn={972-8865-39-2},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the First International Conference on Computer Graphics Theory and Applications - Volume 1: GRAPP,

TI - ALGEBRAIC CURVES IN PARALLEL COORDINATES Avoiding the “Over-Plotting” Problem

SN - 972-8865-39-2

AU - Izhakian Z.

PY - 2006

SP - 150

EP - 157

DO - 10.5220/0001356601500157