Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-28T05:01:05.275Z Has data issue: false hasContentIssue false
  • English
  • Français

How to realize a given number of tangents to four unit balls in ℝ3

Part of: Projective and enumerative geometry

Published online by Cambridge University Press:  26 February 2010

Thorsten Theobald
Thorsten Theobald
Affiliation:
Zentrum Mathematik, Technische Universitat München, Boltzmannstr. 3, D-85747 Garching bei München, Germany. E-mail: [email protected].
Get access

Abstract

By a recent result, the number of common tangent lines to four unit balls in ℝ3 is bounded by 12 unless the four centres are collinear. In the present paper, this result is complemented by showing that indeed every number of tangents k ∈ {0, …, 12} can be established in real space. The constructions combine geometric and algebraic aspects of the tangent problem.

MSC classification

Secondary: 14N10: Enumerative problems (combinatorial problems)
Type
Research Article
Information
Mathematika , Volume 48 , Issue 1-2 , December 2001 , pp. 51 - 62
Copyright
Copyright © University College London 2001

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Agarwal, P. K., Aronov, B. and Sharir, M.. Line transversals of balls and smallest enclosing cylinders in three dimensions. Discrete Compul. Geom., 21 (1999), 373–388.CrossRefGoogle Scholar
2.Bottema, O. and Veldkamp, G. R.. On the lines in space with equal distances to n given points. Geom. Dedicata, 6 (1977), 121–129.CrossRefGoogle Scholar
3.Durand, C.. Symbolic and Numerical Techniques for Constraint Solving. PhD thesis, Purdue University (1998).Google Scholar
4.Harris, J.. Galois groups of enumerative problems. Duke Math. Journal, 46 (1979), 685–724.CrossRefGoogle Scholar
5.Karger, A.. Classical geometry and computers. J. Geom. Graph., 2 (1998), 7–15.Google Scholar
6.Larman, D.. Problem posed in the Problem Session of the DIMACS Workshop on Arrangements, Rutgers University, New Brunswick, NJ (1990).Google Scholar
7.Macdonald, G., Pach, J. and Theobald, T.. Common tangents to four unit balls in R3. Discrete Comput. Geom. 26 (2001), 1–17.CrossRefGoogle Scholar
8.Pedoe, D.. The missing seventh circle. Elem. Math., 25 (1970), 14–15.Google Scholar
9.Salmon, G.. A Treatise on the Analytic Geometry of Three Dimensions, Volume 1. Longmans and Green, London, 7th edition (1928). Reprinted by Chelsea Publishing Company, New York (1958).Google Scholar
10.Schaal, H.. Ein geometrisches Problem der metrischen Getriebesynthese. Sitzungsber., Abt. II, Osterr. Akad. Wiss., 194, (1985), 39–53.Google Scholar
11.Schlafli, L.. On the distributions of surfaces of the third order into species, in reference to the absence or presence of singular points, and the reality of their lines. Phil. Trans. Royal Society of London, 153 (1863), 193–241.Google Scholar
12.Schömer, E., Sellen, J., Teichmann, M. and Yap, C. K.. Efficient algorithms for the smallest enclosing cylinder problem. In Proc. %th Canad. Conf. Comput. Geom., (1996), 264–269.CrossRefGoogle Scholar
13.Segre, B.. The Non-Singular Cubic Surfaces. (Oxford University Press, 1942.)Google Scholar
14.Theobald, T.. An enumerative geometry framework for algorithmic line problems in ℝ3. Siam J. Computing, 31 (2002), 1212–1228.CrossRefGoogle Scholar
15.WeiBbach, B.. Über die senkrechten Projektionen regulärer Simplexe. Beitr. Algebra Geom., 15 (1983), 35–41.Google Scholar