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We show that for $m$ points and $n$ lines in the real plane, the number ofdistinct distances between the points and the lines is$\Omega(m^{1/5}n^{3/5})$, as long as $m^{1/2}\le n\le m^2$. We also prove thatfor any $m$ points in the plane, not all on a line, the number of distancesbetween these points and the lines that they span is $\Omega(m^{4/3})$. Theproblem of bounding the number of distinct point-line distances can be reducedto the problem of bounding the number of tangent pairs among a finite set oflines and a finite set of circles in the plane, and we believe that this latterquestion is of independent interest. In the same vein, we show that $n$ circlesin the plane determine at most $O(n^{3/2})$ points where two or more circlesare tangent, improving the previously best known bound of $O(n^{3/2}\log n)$.Finally, we study three-dimensional versions of the distinct point-linedistances problem, namely, distinct point-line distances and distinctpoint-plane distances. The problems studied in this paper are all new, and thebounds that we derive for them, albeit most likely not tight, are non-trivialto prove. We hope that our work will motivate further studies of these andrelated problems.

Distinct distances between points and lines