Abstract
The envelope of a family of real, rational hypersurfaces is defined by an implicit equation in the parameter space. This equation can be decomposed into factors that are mapped to varieties of different dimension. The factorization can be found using solely gcd computations and polynomial divisions. The decomposition is used to derive some general results about envelopes, which also contribute to the analysis of self-intersections.
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References
Abdel-Malek, K., Yang, J., Blackmore, D., Joy, K.: Swept volumes: fundation, perspectives, and applications. Int. J. Shape Model. 12(1), 87–127 (2006)
Bottema, O., Roth, B.: Theoretical Kinematics. Dover Publications (1990)
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer-Verlag, New York (2007)
Dokken, T., Thomassen, J.: Overview of approximate implicitization. In: Topics in Algebraic Geometry and Geometric Modeling, vol. 334, pp. 169–184. American Mathematical Society (2003)
Flaquer, J., Garate, G., Pargada, M.: Envelopes of moving quadric surfaces. Comput. Aided Geom. Design 9(4), 299–312 (1992)
Kim, Y., Varadhan, G., Lin, M., Manocha, D.: Fast swept volume approximation of complex polyhedral models. Comput. Aided Des. 36(11), 1013–1027 (2004)
Kreyszig, E.: Differential Geometry. Dover (1991)
Peternell, M., Pottmann, H., Steiner, T., Zhao, H.: Swept volumes. Comput. Aided Des. Appl. 2, 599–608 (2005)
Pottmann, H., Peternell, M.: Envelopes-computational theory and applications. In: Spring Conf. on Computer Graphics, pp. 3–23. Comenius Univ., Bratislava (2000)
Rabl, M., Jüttler, B., Gonzalez-Vega, L.: Exact envelope computation for moving surfaces with quadratic support functions. In: Advances in Robot Kinematics: Analysis and Design, pp. 283–290. Springer (2008)
Schulz, T., Jüttler, B.: Envelope computation in the plane by approximate implicitization. Appl. Algebra Engrg. Comm. Comput. 22, 265–288 (2011)
Wenger, P.: A new general formalism for the kinematic analysis of all nonredundant manipulators. In: Robotics and Automation, pp. 442–447. IEEE (1992)
Acknowledgements
The first author was supported by the Marie-Curie Network SAGA (FP7, GA no. 214584), and by the Doctoral Program “Computational Mathematics” (W1214) at Johannes Kepler University, Linz. The authors thank the anonymous referees for their useful comments, in particular for their contributions that led to a correct version of Theorem 1.
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Schulz, T., Jüttler, B. (2012). Decomposing Envelopes of Rational Hypersurfaces. In: Lenarcic, J., Husty, M. (eds) Latest Advances in Robot Kinematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4620-6_24
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DOI: https://doi.org/10.1007/978-94-007-4620-6_24
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-4619-0
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