Common Tangents and Common Transversals

Common Transversals

Given t families, each family consisting of s finite sets, we show that if the families “separate points" in a natural way, and if the union of all the sets in all the families contains more than (s+1)t st 1 1 elements, then a common transversal of the t families exists. In case each family is a covering family, the bound is st st 1. Both of these bounds are best possible. This work extends rec...

Common transversals and tangents to two lines and two quadrics in P3

We solve the following geometric problem, which arises in several threedimensional applications in computational geometry: For which arrangements of two lines and two spheres in R 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 by quadrics...

Common Transversals and Tangents to Two Lines and Two Quadrics in P

We solve the following geometric problem, which arises in several threedimensional applications in computational geometry: For which arrangements of two lines and two spheres in R 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 by quadrics...

Geometric Permutations and Common Transversals

Efficient Algorithms for Common Transversals

Suppose we are gIven a set S of n (possibly intersecting) simple objects in the plane, such that for every pair of objects in S, the intersection of the boundaries of these two objects has at most a connected components. The integer a is independent of n, Le. a.=O (1). \Ve consider the problem of detennining whether there exists a straight line that goes through every object in S. We give an 0 ...