Distinct distances between points and lines

We show that for m points and n lines in R², the number of distinct distances between the points and the lines is Ω(m^1/5n^3/5), as long as m^1/2≤n≤m². We also prove that for any m points in the plane, not all on a line, the number of distances between these points and the lines that they span is Ω(m^4/3). The problem of bounding the number of distinct point-line distances can be reduced to the problem of bounding the number of tangent pairs among a finite set of lines and a finite set of circles in the plane, and we believe that this latter question is of independent interest. In the same vein, we show that n circles in the plane determine at most O(n^3/2) points where two or more circles are tangent, improving the previously best known bound of O(n^3/2log⁡n). Finally, we study three-dimensional versions of the distinct point-line distances problem, namely, distinct point-line distances and distinct point-plane distances. The problems studied in this paper are all new, and the bounds that we derive for them, albeit most likely not tight, are non-trivial to prove. We hope that our work will motivate further studies of these and related problems.