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(m1/5n3/5), as long as m1/2 ≤ n ≤ m2. We also show 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(m4/3). We reduce the problem of bounding the number of distinct point-line distances 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. We believe that this latter question is of independent interest. We also show that n circles in the plane determine at most O(n3/2) points where two or more circles are tangent. This improves the best previously known bound of O(n3/2 log n). Finally, we study the higher-dimensional version of the distinct point-line distances, namely, the number of distinct point-hyperplane distances in Rd