Common transversals and tangents to two lines and two quadrics in P^3

We solve the following geometric problem, which arises in several three-dimensional applications in computational geometry: For which arrangements of two lines and two spheres in 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 R^3 by quadrics in projective space 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 2 lines and 2 quadrics have infinitely many transversals and tangents: In the nine-dimensional projective space P^9 of quadrics, this is a curve of degree 24 consisting of 12 plane conics, a remarkably reducible variety.Comment: 26 pages, 9 .eps figures, web page with more pictures and and archive of computations: http://www.math.umass.edu/~sottile/pages/2l2s

Real k-flats tangent to quadrics in R^n

Let d_{k,n} and #_{k,n} denote the dimension and the degree of the Grassmannian G_{k,n} of k-planes in projective n-space, respectively. For each k between 1 and n-2 there are 2^{d_{k,n}} \cdot #_{k,n} (a priori complex) k-planes in P^n tangent to d_{k,n} general quadratic hypersurfaces in P^n. We show that this class of enumerative problem is fully real, i.e., for each k between 1 and n-2 there exists a configuration of d_{k,n} real quadrics in (affine) real space R^n so that all the mutually tangent k-flats are real.Comment: 10 pages, 3 figures. Minor revisions, to appear in Proc. AM

Lines Tangent to 2n-2 spheres in R^n

We show that there are 3 \cdot 2^(n-1) complex common tangent lines to 2n-2 general spheres in R^n and that there is a choice of spheres with all common tangents real.Comment: Minor revisions. Trans. AMer. Math. Soc., to appear. 15 pages, 3 .eps figures; also a web page with computer code verifying the computations in the paper and with additional picture

Common Tangents to Spheres in R3R3

Article dans revue scientifique avec comité de lecture. internationale.International audienceWe prove that four spheres in $R3$ have infinitely many real common tangents if and only if they have aligned centers and at least one real common tangent

Transversals to Line Segments in Three-Dimensional Space

We completely describe the structure of the connected components of transversals to a collection of n line segments in R3. We show that n \u3e 3 arbitrary line segments in R3 admit 0, 1, . . . , n or infinitely many line transversals. In the latter case, the transversals form up to n connected components

Contributions to the theory of apolarity

“I am the holder of a Scholarship from the University of Madras and one from the Madras Government. Also of a grant from the British Department of Scientific Research.” -- Caree