Abstract
This paper deals with the decision problem of the surjectivity of a rational surface parametrization. We give sufficient conditions for a parametrization to be surjective, and we describe different families of parametrizations that satisfy these criteria. In addition, we consider the problem of computing a superset of the points not covered by the parametrization. In this context, we report on the case of parametrizations without projective base points and we analyze the particular case of rational ruled surfaces.
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References
Andradas, C., Recio, T., Sendra, J.R., Tabera, L.F.: On the simplification of the coefficients of a parametrization. J. Symbolic Comput. 44(2), 192–210 (2009)
Andradas, C., Recio, T., Sendra, J.R., Tabera, L.F., Villarino, C.: Proper real reparametrization of rational ruled surfaces. Comput. Aided Geom. Design 28(2), 102–113 (2011)
Busé, L., Cox, D., D’Andrea, C.: Implicitization of surfaces in \(\mathbb{P}^3\) in the presence of base points. J. Algebra Appl. 2(2), 189–214 (2003)
Bodnár, G., Hauser, H., Schicho, J., Villamayor, O.: Plain varieties. Bull. Lond. Math. Soc. 40(6), 965–971 (2008)
Bajaj, C.L., Royappa, A.V.: Finite representations of real parametric curves and surfaces. Internat. J. Comput. Geom. Appl. 5(3), 313–326 (1995)
Chen, F.: Reparametrization of a rational ruled surface using the \(\mu \)-basis. Comput. Aided Geom. Design 20(1), 11–17 (2003)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. Undergraduate Texts in Mathematics, 3rd edn. Springer, New York (2007)
Farin, G., Hoschek, J., Kim, M.-S. (eds.): Handbook of Computer Aided Geometric Design. North-Holland, Amsterdam (2002)
Hoschek, J., Lasser, D.: Fundamentals of Computer Aided Geometric Design. A. K. Peters Ltd., Wellesley (1993). Translated from the 1992 German edition by L.L. Schumaker
Pérez-Díaz, S., Shen, L.-Y.: Characterization of rational ruled surfaces. J. Symbolic Comput. 50, 450–464 (2013)
Pérez-Díaz, S., Sendra, J.R., Villarino, C.: A first approach towards normal parametrizations of algebraic surfaces. Internat. J. Algebra Comput. 20(8), 977–990 (2010)
Recio, T., Sendra, J.R., Tabera, L.F., Villarino, C.: Generalizing circles over algebraic extensions. Math. Comp. 79(270), 1067–1089 (2010)
Rafael Sendra, J.: Normal parametrizations of algebraic plane curves. J. Symbolic Comput. 33(6), 863–885 (2002)
Saito, T., Sederberg, T.W.: Rational-ruled surfaces: implicitization and section curves. Graph. Models Image Process. 57(4), 334–342 (1995)
Sendra, J.R., Sevilla, D., Villarino, C.: Covering of surfaces parametrized without projective base points. In: Proceedings of ISSAC 2014, pp. 375–380. ACM Press (2014)
Sendra, J.R., Sevilla, D., Villarino, C.: Covering rational ruled surfaces. arXiv:1406.2140v1 [math.AG]
Acknowledgements
This work was developed, and partially supported, by the Spanish Ministerio de Economía y Competitividad under Project MTM2011-25816-C02-01. The first and third authors are members of the Research Group ASYNACS (Ref. CCEE2011/R34).
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Sendra, J.R., Sevilla, D., Villarino, C. (2015). Some Results on the Surjectivity of Surface Parametrizations. In: Gutierrez, J., Schicho, J., Weimann, M. (eds) Computer Algebra and Polynomials. Lecture Notes in Computer Science(), vol 8942. Springer, Cham. https://doi.org/10.1007/978-3-319-15081-9_11
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DOI: https://doi.org/10.1007/978-3-319-15081-9_11
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