Christian Stussak
ABSTRACT
In this work we propose an algorithm for the exact rasterization of a given real algebraic plane curve, which is the set of real solutions of a bivariate polynomial equation F(x, y) = 0. Our algorithm first divides the image plane into simple rectangles, where the curve has no local extreme values. In these blocks the topology is known and the direction of the curve can easily be determined. Subsequent we efficiently trace the curve from one row of pixels to the next by using either tests for a sign changes of F(x, y) in simple cases or real root counting via Sturm-Habicht sequences in presence of dense curve arcs. In contrast to other approaches, the curve tracing is performed at the given resolution and will never subdivided the image plane below pixel level to obtain a correct result.
- Alberti, L., and Mourrain, B. 2007. Visualisation of implicit algebraic curves. Computer Graphics and Applications, Pacific Conference on 0, 303--312. Google Scholar
Digital Library
- Cheng, J.-S., Gao, X.-S., and Li, J. 2009. Root isolation for bivariate polynomial systems with local generic position method. In Proceedings of the 2009 international symposium on Symbolic and algebraic computation, ACM, New York, NY, USA, ISSAC '09, 103--110. Google ScholarDigital Library
- Eigenwillig, A., Kerber, M., and Wolpert, N. 2007. Fast and exact geometric analysis of real algebraic plane curves. In Proceedings of the 2007 international symposium on Symbolic and algebraic computation, ACM, New York, NY, USA, ISSAC '07, 151--158. Google ScholarDigital Library
- Emeliyanenko, P., Berberich, E., and Sagraloff, M. 2009. Visualizing arcs of implicit algebraic curves, exactly and fast. In Advances in Visual Computing, vol. 5875 of Lecture Notes in Computer Science. Springer Berlin/Heidelberg, 608--619. 10.1007/978-3-642-10331-5_57. Google ScholarDigital Library
- Falai, C., Yuyu, F., and Kozak, J. 1997. Tracing a planar algebraic curve. Applied Mathematics - A Journal of Chinese Universities 12, 15--24. 10.1007/s11766-997-0002-2.Google ScholarCross Ref
- González-Vega, L., Recio, T., and Lombardi, H. 1998. Sturm-Habicht sequence, determinants and real roots of univariate polynomials. 300--316.Google Scholar
- Labs, O. 2010. A List of Challenges for Real Algebraic Plane Curve Visualization Software. Nonlinear Computational Geometry, 137--164.Google Scholar
- Lickteig, T., and Roy, M.-F. 2001. Sylvester-Habicht sequences and fast cauchy index computation. Journal of Symbolic Computation 31, 3, 315--341. Google ScholarDigital Library
- Martin, R., Shou, H., Voiculescu, I., Bowyer, A., and Wang, G. 2002. Comparison of interval methods for plotting algebraic curves. Computer Aided Geometric Design 19, 7, 553--587. Google ScholarDigital Library
- Seidel, R., and Wolpert, N. 2005. On the exact computation of the topology of real algebraic curves. In Proceedings of the twenty-first annual symposium on Computational geometry, ACM, New York, NY, USA, SCG '05, 107--115. Google ScholarDigital Library
- Stussak, C., and Schenzel, P. 2011. Parallel computation of bivariate polynomial resultants on graphics processing units. In Proceedings of the Para 2010 conference, Springer. To appear. Google ScholarDigital Library
- Stussak, C., 2011. An algorithm for the exact rasterization of real algebraic plane curves. In preparation.Google Scholar
- Taubin, G. 1993. An accurate algorithm for rasterizing algebraic curves. In Proceedings on the second ACM symposium on Solid modeling and applications, ACM, New York, NY, USA, SMA '93, 221--230. Google ScholarDigital Library
Index Terms
- On exact rasterization of real algebraic plane curves
Recommendations
Rational Quadratic Approximation to Real Plane Algebraic Curves
GMP '04: Proceedings of the Geometric Modeling and Processing 2004An algorithm is proposed to give a global approximationto an implicit real plane algebraic curve with rationalquadratic B-splines. The algorithm consists of three steps:curve segmentation, segment approximation and curve tracing.The curve is first ...
Algorithms for Rational Real Algebraic Curves
In this paper, we study fundamental properties of real curves, especially of rational real curves, and we derive several algorithms to decide the reality and rationality of curves in the complex plane. Furthermore, if the curve is real and rational, we ...
Algebro-geometric analysis of bisectors of two algebraic plane curves
In this paper, a general theoretical study, from the perspective of the algebraic geometry, of the untrimmed bisector of two real algebraic plane curves is presented. The curves are considered in C 2 , and the real bisector is obtained by restriction to ...
Comments