Algebraic methods for computing smallest enclosing and circumscribing cylinders of simplices

Applicable Algebra in Engineering, Communications and Computing

Volumn 14, Issue 6, 2004, Pages 439-460

  Brandenberg, René ^a   Theobald, Thorsten ^a  

^a TECHNICAL UNIVERSITY OF MUNICH   (Germany)

Author keywords

Circumscribing cylinder; Outer radius; Polynomial equations; Simplex; Smallest enclosing cylinder

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; COMPUTATIONAL GEOMETRY; COMPUTER SOFTWARE; COMPUTER VISION; EQUIVALENCE CLASSES; LAGRANGE MULTIPLIERS; ROBOTICS; SET THEORY; SPHERES; THEOREM PROVING; VECTORS;

CIRCUMSCRIBING CYLINDERS; ELEMENTARY INVARIANT THEORY; EUCLIDEAN SPACE; SIMPLICES;

POLYNOMIALS;

EID: 1842482360     PISSN: 09381279     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00200-003-0146-0     Document Type: Article

References (32)

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)
2. Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper. Springer-Verlag, Berlin, 1934
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
4. Brandenberg, R.: Radii of regular polytopes. Preprint, 2003. math.GM/0308121
5. Brandenberg, R., Theobald, T.: Radii minimal projections of simplices and constrained optimization of symmetric polynomials. Preprint, 2003. math.MG/0311017
6. Chan, T.M.: Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus. Internat. J. Comp. Geom. Appl. 12, 67-85 (2002)
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 edition
8. Cox, D., Little, J., O'Shea, D.: Using Algebraic Geometry. Graduate Texts in Mathematics, 185, Springer-Verlag, New York, 1998
9. Coxeter, H.S.M.: Introduction to Geometry. John Wiley & Sons, 1961
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)
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, 2002
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)
13. Emiris, I., Canny, J.: Efficient incremental algorithms for the sparse resultant and the mixed volume. J. Symb. Comp. 20, 117-149 (1995)
14. Graham, R.L., Knuth, D.E., Patashnik, O.: Concrete mathematics. Addison-Wesley, 1989
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
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)
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)
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, 1997
19. Har-Peled, S., Varadarajan, K.: Projective clustering in high dimensions using core sets. Proc. ACM Symposium on Computational Geometry '02 (Barcelona), 312-318, 2002
20. Hilbert, D., Cohn-Vossen, S.: Anschauliche Geometrie. Springer-Verlag, Berlin, 1932. Translation: Geometry and the Imagination. Chelsea Publ., New York, 1952
21. Kupitz, Y.S., Martini, H.: Equifacial tetrahedra and a famous location problem. Math. Gazette 83, 464-467 (1999)
22. 3. Discrete Comput. Geom. 26, 1-17 (2001)
23. Schaal, H.: Ein geometrisches Problem der metrischen Getriebesynthese. In Sitzungsber., Abt. II, Österr. Akad. Wiss. 194, 39-53 (1985)
24. Schömer, E., Sellen, J., Teichmann, M., Yap, C.: Smallest enclosing cylinders. Algorithmica 27, 170-186 (2000)
25. Stanley, R.P.: Enumerative Combinatorics. Cambridge University Press, 1997
26. n. Trans. Amer. Math. Soc. 354, 4815-4829 (2002)
27. Sturmfels, B.: Solving Systems of Polynomial Equations. CBMS series, 97, AMS, Providence, 2002
28. Sturmfels, B.: Algorithms in Invariant Theory. RISC Series in Symbolic Computation, Springer-Verlag, Wien, 1993
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)
30. Verschelde, J.: PHCpack: A general-purpose solver for polynomial systems by homotopy continuation. ACM Trans. Math. Software 25, 251-276 (1999)
31. Weißbach, B.: Über die senkrechten Projektionen regulärer Simplexe. Beitr. Algebra Geom. 15, 35-41 (1983)
32. Weißbach, B.: Über Umkugeln von Projektionen regulärer Simplexe. Beitr. Algebra Geom. 16, 127-137 (1983)