Lines tangent to 2𝑛-2 spheres in ℝⁿ

Sottile, Frank; Theobald, Thorsten

doi:10.1090/S0002-9947-02-03014-3

Lines tangent to $2n-2$ spheres in ${\mathbb R}^n$
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by Frank Sottile and Thorsten Theobald PDF

Trans. Amer. Math. Soc. 354 (2002), 4815-4829 Request permission

Abstract:

We show that for $n \ge 3$ there are $3 \cdot 2^{n-1}$ complex common tangent lines to $2n-2$ general spheres in $\mathbb {R}^n$ and that there is a choice of spheres with all common tangents real.

References

P. K. Agarwal, B. Aronov, and M. Sharir, Line transversals of balls and smallest enclosing cylinders in three dimensions, Discrete Comput. Geom. 21 (1999), no. 3, 373–388. MR 1672980, DOI 10.1007/PL00009427
P. Aluffi and W. Fulton, Lines tangent to four surfaces containing a curve, in preparation.
T.M. Chan, Approximating the diameter, width, smallest enclosing cylinder, and minimum-width annulus, Proc. ACM Symposium on Computational Geometry (Clear Water Bay, Hong Kong), 2000, pp. 300–309.
M. Chasles, Construction des coniques qui satisfont à cinque conditions, C. R. Acad. Sci. Paris 58 (1864), 297–308.
U. Faigle, W. Kern, and M. Streng, Note on the computational complexity of $j$-radii of polytopes in $\textbf {R}^n$, Math. Programming 73 (1996), no. 1, Ser. A, 1–5. MR 1389665, DOI 10.1016/0025-5610(95)00036-4
William Fulton, Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 2, Springer-Verlag, Berlin, 1984. MR 732620, DOI 10.1007/978-3-662-02421-8
Alfred Rosenblatt, Sur les points singuliers des équations différentielles, C. R. Acad. Sci. Paris 209 (1939), 10–11 (French). MR 85
S. L. Kleiman and Dan Laksov, Schubert calculus, Amer. Math. Monthly 79 (1972), 1061–1082. MR 323796, DOI 10.2307/2317421
I.G. Macdonald, J. Pach, and T. Theobald, Common tangents to four unit balls in $\mathbb {R}^3$, Discrete Comput. Geom. 26:1 (2001), 1–17.
G. Megyesi, Lines tangent to four unit spheres with coplanar centres, Discrete Comput. Geom. 26:4 (2001), 493–497.
G. Megyesi, Configurations of $2n-2$ quadrics in $\mathbb {R}^n$ with $3\cdot 2^{n-1}$ common tangent lines, 2001.
Felice Ronga, Alberto Tognoli, and Thierry Vust, The number of conics tangent to five given conics: the real case, Rev. Mat. Univ. Complut. Madrid 10 (1997), no. 2, 391–421. MR 1605670
Hermann Schaal, Ein geometrisches Problem der metrischen Getriebesynthese, Österreich. Akad. Wiss. Math.-Natur. Kl. Sitzungsber. II 194 (1985), no. 1-3, 39–53 (German). MR 833890
E. Schömer, J. Sellen, M. Teichmann, and Chee Yap, Smallest enclosing cylinders, Algorithmica 27 (2000), no. 2, 170–186. MR 1746779, DOI 10.1007/s004530010011
Frank Sottile, Enumerative geometry for real varieties, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 435–447. MR 1492531, DOI 10.1016/s0022-4049(97)00029-7
—, From enumerative geometry to solving systems of polynomial equations with Macaulay 2, in Computations in Algebraic Geometry with Macaulay 2, ed. by D. Eisenbud, D. Grayson, M. Stillman, and B. Sturmfels, Algorithms and Computations in Mathematics, vol. 8, Springer-Verlag, Berlin, 2001, pp. 101–129.
—, Enumerative real algebraic geometry, in preparation for Proceedings of the DIMACS workshop on Algorithmic and Quantitative Aspects of Real Algebraic Geometry in Mathematics and Computer Science, Ed. by S. Basu and L. Gonzalez-Vega, for DIMACS book series, Amer. Math. Soc., Providence, RI, 2001.
F. Sottile and T. Theobald, Real lines tangent to $2n-2$ quadrics in $\mathbb {R}^n$, 2002.
J. Steiner, Elementare Lösung einer geometrischen Aufgabe, und über einige damit in Beziehung stehende Eigenschaften der Kegelschnitte, J. Reine Angew. Math. 37 (1848), 161–192.