Abstract
Recently, the visualization of implicitly given algebraic curves and surfaces has become an area of active research. Most of the approaches either use raytracing, subdivision or sweeping techniques to produce a good approximate picture of the varieties, sometimes by using hardware equipment such as graphics processing units.
We provide a list of equations of plane curves which may serve as a list of benchmarks for visualization software. In most cases, we give whole series of examples which yield equations for infinitely many degrees. Even for low degrees, there is currently no software which visualizes all examples correctly in real–time, so we call them challenges.
For most of the equations in our list, we are able to prove that they are at least close to the most difficult possible ones. For convenience, our list is also available in the form of a text file. Moreoever, the paper includes a brief introduction to some of the terminology from singularity theory for researchers from the computer graphics community because singularities appear frequently when treating complicated cases.
Keywords
- Real algebraic geometry
- computational geometry
- geometric modeling
- plane curves
- singularities
- visualization
- algorithms
- benchmarks
- challenges
AMS(MOS) subject classifications
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
V.I. Arnold, S.M. Gusein-Zade, and A.N. Varchenko, Singularities of differentiable maps, Birkhäuser, 1985 (two volumes).
V.I. Arnold, Singularity Theory, London Math. Soc. Lecture Note Series, Vol. 53, Cambridge University Press, 1981.
J. Bochnak, M. Coste, and M.-F. Roy, Real algebraic geometry, Springer, 1998.
E. Brieskorn and H. Knörrer, Plane Algebraic Curves, Birkhäuser, 1986.
R. Benedetti and J.J. Risler, Real algebraic and semi-algebraic sets, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990. MR 1070358 (91j:14045)
M. Coste, Épaississement d'une hypersurface algébrique réelle, Proc. Japan Acad. Ser. A Math. Sci. 68 (1992), no. 7, 175–180. MR 1193176 (94b:14056)
[Dim87] A. Dimca, Topics on Real and Complex Singularities, Vieweg, 1987.
A. Dickenstein, J.M. Rojas, K. Rusek, and J. Shih, Extremal Real Algebraic Geometry and A-Discriminants, Moscow Mathematical Journal 7 (2007), no. 3, 425–452.
A.H. Durfee, Fifteen characterizations of rational double points and simple critical points, Enseign. Math.,II. Sér. 25 (1979), 132–163.
[GPS06] G.-M. Greuel, G. Pfister, and H. Schönemann, Singular 3.0, A Computer Algebra System for Polynomial Computations, Centre for Computer Algebra, Univ. Kaiserslautern, 2006, http://www.singular.uni-kl.de.
S.M. Gusein-Zade and N.N. Nekhoroshev, On singularities of type A k on simple curves of fixed degree, Funct. Anal. Appl. 34 (2000), 214–215.
A. Harnack, Ueber die Vieltheiligkeit der ebenen algebraischen Curven, Math. Ann. 10 (1876), no. 2, 189–198. MR 1509883
R. Kenyon and A. Okounkov, Planar dimers and Harnack curves, Duke Math. J. 131 (2006), no. 3, 499–524. MR 2219249
[Lab03] O. Labs, Algebraic Surface Homepage. Information, Images and Tools on Algebraic Surfaces, www.AlgebraicSurface.net, 2003.
E. Shustin, Gluing of singular and critical points, Topology 37 (1998), no. 1, 195–217.
A.N. Varchenko, On the Semicontinuity of the Spectrum and an Upper Bound for the Number of Singular Points of a Projective Hypersurface, J. Soviet Math. 270 (1983), 735–739.
E. Westenberger, Real hypersurfaces with many simple singularities, Rev. Mat. Complut. 18 (2005), no. 2, 455–464.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag New York
About this paper
Cite this paper
Labs, O. (2009). A List of Challenges for Real Algebraic Plane Curve Visualization Software. In: Emiris, I., Sottile, F., Theobald, T. (eds) Nonlinear Computational Geometry. The IMA Volumes in Mathematics and its Applications, vol 151. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-0999-2_6
Download citation
DOI: https://doi.org/10.1007/978-1-4419-0999-2_6
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-0998-5
Online ISBN: 978-1-4419-0999-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)