Finding stabbing numbers for lines for a set of segments in the plane
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  H. Edelsbrunner, H. A. Maurer, F. P. Preparata, A. L. Rosenberg, E. Welzl, D. Wood.
  "Stabbing line segments."
  BIT Numerical Mathematics.
  1982, Volume 22, Issue 3, pp 274-281.
  (Springer link.)


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One can use a segment tree data structure, or an interval tree.
Segment trees offer $O(\log n + k)$ query time, where $k$ is the number of segments intersected. Space complexity is $O(n \log n)$.