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2006.332: Common tranversals and tangents to two lines and two quadrics in P

2006.332: Gàbor Megyesi, Frank Sottile and Thorsten Theobald (2003) Common tranversals and tangents to two lines and two quadrics in P. Discrete and Computational Geometry, 30 (4). pp. 543-571. ISSN 1432-0444

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DOI: 10.1007/s00454-003-0789-4


We solve the following geometric problem, which arises in several {\mbox three-dimensional} applications in computational geometry: For which arrangements of two lines and two spheres in ${\Bbb R}^3$ are there infinitely many lines simultaneously transversal to the two lines and tangent to the two spheres? We also treat a generalization of this problem to projective quadrics. Replacing the spheres in ${\bf R}^3$ by quadrics in projective space ${\Bbb P}^3$, and fixing the lines and one general quadric, we give the following complete geometric description of the set of (second) quadrics for which the two lines and two quadrics have infinitely many transversals and tangents: in the nine-dimensional projective space ${\Bbb P}^9$ of quadrics, this is a curve of degree 24 consisting of 12 plane conics, a remarkably reducible variety.

Item Type:Article
Subjects:MSC 2000 > 51 Geometry (See also algebraic geometry)
MSC 2000 > 53 Differential geometry
MIMS number:2006.332
Deposited By:Miss Louise Stait
Deposited On:17 August 2006

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