Distinct distances between points and lines

We show that for m points and n lines in R 2 R 2 , the number of distinct distances between the points and the lines is Ω(m 1/5 n 3/5 ) Ω ( m 1 / 5 n 3 / 5 ) , as long as m 1/2 ≤n≤m 2 m 1 / 2 ≤ n ≤ m 2 . 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 ) 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 ) O ( n 3 / 2 ) points where two or more circles are tangent, improving the previously best known bound of O(n 3/2 log⁡n) O ( n 3 / 2 log ⁡ 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.

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