Abstract
We solve in ℝn the problem of distance evaluation between a quadric and a manifold obtained as the intersection of another quadric and a linear manifold. Application of Elimination Theory algorithms for the system of algebraic equations of the Lagrange multipliers method results in construction of the distance equation, i.e., a univariate algebraic equation one of the zeros of which (generically minimal positive) coincides with the square of the distance between considered manifolds. We also deduce the necessary and sufficient algebraic conditions under which the manifolds intersect and propose an algorithm for finding the coordinates of their nearest points.
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References
Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms. Springer, New York (2007)
Gelfand, I.M., Kapranov, M.M., Zelevinsky, A.V.: Discriminants, Resultants and Multidimensional Determinants. Springer, New York (1994)
Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press (1986)
Kalinina, E.A., Uteshev, A.Y.: Determination of the number of roots of a polynominal lying in a given algebraic domain. Linear Algebra Appl. 185, 61–81 (1993)
Netto, E.: Rationale funktionen einer veränderlichen; ihre nullstellen. In: Encyklopädie der Mathematischen Wissenschaften, vol. I, Part 1, pp. 227–253. Teubner, Leipzig (1898)
Perron, O.: Algebra, vol. 1. De Gruyter, Berlin (1927)
Schneider, P.J., Eberly, D.H.: Geometric Tools for Computer Graphics. Elsevier, San Francisco (2003)
Uteshev, A.Y., Yashina, M.V.: Distance computation from an ellipsoid to a linear or a quadric surface in IRn. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 392–401. Springer, Heidelberg (2007)
Uteshev, A.Y., Yashina, M.V.: Metric problems for quadrics in multidimensional space. J. Symb. Comput. 68, 287–315 (2015)
Wang, W., Krasauskas, R.: Interference analysis of conics and quadrics. Contemp. Mathematics 334, 25–36 (2003)
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© 2015 Springer International Publishing Switzerland
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Uteshev, A.Y., Yashina, M.V. (2015). Distance Evaluation Between an Ellipse and an Ellipsoid. In: Gerdt, V., Koepf, W., Seiler, W., Vorozhtsov, E. (eds) Computer Algebra in Scientific Computing. CASC 2015. Lecture Notes in Computer Science(), vol 9301. Springer, Cham. https://doi.org/10.1007/978-3-319-24021-3_34
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DOI: https://doi.org/10.1007/978-3-319-24021-3_34
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