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Algebraic Methods for Computing Smallest Enclosing and Circumscribing Cylinders of Simplices

  • René Brandenberg
  • Thorsten Theobald
Article

Abstract.

We provide an algebraic framework to compute smallest enclosing and smallest circumscribing cylinders of simplices in Euclidean space n . Explicitly, the computation of a smallest enclosing cylinder in 3 is reduced to the computation of a smallest circumscribing cylinder. We improve existing polynomial formulations to compute the locally extreme circumscribing cylinders in 3 and exhibit subclasses of simplices where the algebraic degrees can be further reduced. Moreover, we generalize these efficient formulations to the n-dimensional case and provide bounds on the number of local extrema. Using elementary invariant theory, we prove structural results on the direction vectors of any locally extreme circumscribing cylinder for regular simplices.

Keywords

Smallest enclosing cylinder Circumscribing cylinder Simplex Outer radius Polynomial equations 

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References

  1. 1.
    Agarwal, P.K., Aronov, B., Sharir, M.: Line transversals of balls and smallest enclosing cylinders in three dimensions. Discrete Comput. Geom. 21, 373–388 (1999)MathSciNetMATHGoogle Scholar
  2. 2.
    Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper. Springer-Verlag, Berlin, 1934Google Scholar
  3. 3.
    Brandenberg, R.: Radii of Convex Bodies. Ph.D. thesis, Dept. of Mathematics, Technische Universität München, 2002. http://tumb1.biblio.tu-muenchen.de/publ/diss/ ma/2002/brandenberg.html~Google Scholar
  4. 4.
    Brandenberg, R.: Radii of regular polytopes. Preprint, 2003. math.GM/0308121Google Scholar
  5. 5.
    Brandenberg, R., Theobald, T.: Radii minimal projections of simplices and constrained optimization of symmetric polynomials. Preprint, 2003. math.MG/0311017Google Scholar
  6. 6.
    Chan, T.M.: Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus. Internat. J. Comp. Geom. Appl. 12, 67–85 (2002)CrossRefMATHGoogle Scholar
  7. 7.
    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra. UTM, Springer-Verlag, New York, 1996, second editionGoogle Scholar
  8. 8.
    Cox, D., Little, J., O’Shea, D.: Using Algebraic Geometry. Graduate Texts in Mathematics, 185, Springer-Verlag, New York, 1998Google Scholar
  9. 9.
    Coxeter, H.S.M.: Introduction to Geometry. John Wiley & Sons, 1961Google Scholar
  10. 10.
    Devillers, O., Mourrain, B., Preparata, F.P., Trébuchet, Ph.: On circular cylinders through four or five points in space. Discrete Comput. Geom. 29, 83–104 (2002)CrossRefMATHGoogle Scholar
  11. 11.
    Dos Reis, G., Mourrain, B., Rouillier, F., Trébuchet, Ph.: SYNAPS: An environment for symbolic and numeric computation. Proc. International Congress of Mathematical Software 2002, Beijing, 239-249, World Scientific, 2002Google Scholar
  12. 12.
    Eggleston, H.G.: Notes on Minkowski geometry (I): Relations between the circumradius, diameter, inradius, and minimal width of a convex set. J. London Math. Soc. 33, 76–81 (1958)MATHGoogle Scholar
  13. 13.
    Emiris, I., Canny, J.: Efficient incremental algorithms for the sparse resultant and the mixed volume. J. Symb. Comp. 20, 117–149 (1995)CrossRefMathSciNetMATHGoogle Scholar
  14. 14.
    Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete mathematics. Addison-Wesley, 1989Google Scholar
  15. 15.
    Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 2.0. A computer algebra system for polynomial computations. Centre for Computer Algebra, University of Kaiserslautern, 2001. http://www.singular.uni-kl.de~Google Scholar
  16. 16.
    Gritzmann, P., Klee, V.: Inner and outer j-radii of convex bodies in finite-dimensional normed spaces. Discrete Comput. Geom. 7, 255–280 (1992)MathSciNetMATHGoogle Scholar
  17. 17.
    Gritzmann, P., Klee, V.: Computational complexity of inner and outer j-radii of polytopes in finite-dimensional normed spaces. Math. Program. 59A, 163–213 (1993)MathSciNetMATHGoogle Scholar
  18. 18.
    Gritzmann, P., Klee, V.: Computational convexity. In Handbook of Discrete and Computational Geometry (J.E. Goodman, J. O’Rourke, eds.), 491–515, CRC Press, Boca Raton, 1997Google Scholar
  19. 19.
    Har-Peled, S., Varadarajan, K.: Projective clustering in high dimensions using core sets. Proc. ACM Symposium on Computational Geometry ‘02 (Barcelona), 312–318, 2002Google Scholar
  20. 20.
    Hilbert, D., Cohn-Vossen, S.: Anschauliche Geometrie. Springer-Verlag, Berlin, 1932. Translation: Geometry and the Imagination. Chelsea Publ., New York, 1952Google Scholar
  21. 21.
    Kupitz, Y.S., Martini, H.: Equifacial tetrahedra and a famous location problem. Math. Gazette 83, 464–467 (1999)MATHGoogle Scholar
  22. 22.
    Macdonald, I.G., Pach, J., Theobald, T.: Common tangents to four unit balls in ℝ3. Discrete Comput. Geom. 26, 1–17 (2001)MathSciNetMATHGoogle Scholar
  23. 23.
    Schaal, H.: Ein geometrisches Problem der metrischen Getriebesynthese. In Sitzungsber., Abt. II, Österr. Akad. Wiss. 194, 39–53 (1985)Google Scholar
  24. 24.
    Schömer, E., Sellen, J., Teichmann, M., Yap, C.: Smallest enclosing cylinders. Algorithmica 27, 170–186 (2000)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Stanley, R.P.: Enumerative Combinatorics. Cambridge University Press, 1997Google Scholar
  26. 26.
    Sottile, F., Theobald, T.: Lines tangent to 2n-2 spheres in ℝn. Trans. Amer. Math. Soc. 354, 4815–4829 (2002)CrossRefMathSciNetMATHGoogle Scholar
  27. 27.
    Sturmfels, B.: Solving Systems of Polynomial Equations. CBMS series, 97, AMS, Providence, 2002Google Scholar
  28. 28.
    Sturmfels, B.: Algorithms in Invariant Theory. RISC Series in Symbolic Computation, Springer-Verlag, Wien, 1993Google Scholar
  29. 29.
    Theobald, T.: Visibility computations: From discrete algorithms to real algebraic geometry. In S. Basu and L. Gonzalez-Vega (eds.), Algorithmic and Quantitative Real Algebraic Geometry in Mathematics and Computer Science, AMS DIMACS series, 60, 207–219 (2003)Google Scholar
  30. 30.
    Verschelde, J.: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Software 25, 251–276 (1999)CrossRefMATHGoogle Scholar
  31. 31.
    Weißbach, B.: Über die senkrechten Projektionen regulärer Simplexe. Beitr. Algebra Geom. 15, 35–41 (1983)Google Scholar
  32. 32.
    Weißbach, B.: Über Umkugeln von Projektionen regulärer Simplexe. Beitr. Algebra Geom. 16, 127–137 (1983)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Zentrum MathematikTechnische Universität MünchenGarching beiMünchen

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