Abstract
Consider a rational family of planar rational curves in a certain region of interest. We are interested in finding an approximation to the implicit representation of the envelope. Since exact implicitization methods tend to be very costly, we employ an adaptation of approximate implicitization to envelope computation. Moreover, by utilizing an orthogonal basis in the construction process, the computational times can be shortened and the numerical condition improved. We provide an example to illustrate the performance of our approach.
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References
Abdel-Malek, K., Yang, J., Blackmore, D., Joy, K.: Swept volumes: Foundation, perspectives, and applications. Int. J. Shape Model. 12(1), 87–127 (2006)
Barrowclough, O., Dokken, T.: Approximate implicitization using linear algebra. J. Appl. Math. (2012). doi:10.1155/2012/293746
Dokken, T.: Approximate implicitization. In: Mathematical Methods for Curves and Surfaces, Oslo 2000, pp. 81–102 (2001)
Dokken, T., Thomassen, J.: Weak approximate implicitization. In: IEEE International Conference on Shape Modeling and Applications, SMI 2006, 2006, pp. 204–214 (2006)
Hoffmann, C.: Implicit curves and surfaces in CAGD. IEEE Comput. Graph. Appl. 13(1), 79–88 (1993)
Kim, Y., Varadhan, G., Lin, M., Manocha, D.: Fast swept volume approximation of complex polyhedral models. Comput. Aided Des. 36(11), 1013–1027 (2004)
Peternell, M., Pottmann, H., Steiner, T., Zhao, H.: Swept volumes. Comput-Aided Des. Appl. 2, 599–608 (2005)
Rabl, M., Jüttler, B., Gonzalez-Vega, L.: Exact envelope computation for moving surfaces with quadratic support functions. In: Lenarčič, J., Wenger, P. (eds.) Adv. 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 Eng. Commun. Comput. 22, 265–288 (2011)
Sendra, J., Winkler, F., Perez-Diaz, S.: Rational Algebraic Curves. Springer (2007)
Trefethen, L.N.: Spectral Methods in MATLAB. SIAM, Philadelphia, USA (2000)
Acknowledgements
The research leading to these results has received funding from the European Community’s Seventh Framework Programme FP7/2007-2013 under grant agreement n° PITN-GA-2008-214584 (SAGA), and from the Research Council of Norway (IS-TOPP). It was also supported by the Doctoral Program “Computational Mathematics” (W1214) at the Johannes Kepler University of Linz.
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Barrowclough, O.J.D., Jüttler, B., Schulz, T. (2012). Fast Approximate Implicitization of Envelope Curves Using Chebyshev Polynomials. In: Lenarcic, J., Husty, M. (eds) Latest Advances in Robot Kinematics. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-4620-6_26
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DOI: https://doi.org/10.1007/978-94-007-4620-6_26
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-007-4619-0
Online ISBN: 978-94-007-4620-6
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