An Enumerative Geometry Framework for Algorithmic Line Problems in $\mathbb R^3$

We investigate the enumerative geometry aspects of algorithmic line problems when the admissible bodies are balls or polytopes. For this purpose, we study the common tangent lines/transversals to k balls of arbitrary radii and 4− k lines in R3. In particular, we compute tight upper bounds for the maximum number of real common tangents/transversals in these cases. Our results extend the results of Macdonald, Pach, and Theobald who investigated common tangents to four unit balls in R3 [Discrete Comput. Geom., 26 (2001), pp. 1–17].