## Abstract

We show that for m points and n lines in R^{2}, the number of distinct distances between the points and the lines is Ω(m^{1/5}n^{3/5}), as long as 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}). 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^{3/2}logn). 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.

Original language | English |
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Pages (from-to) | 2-15 |

Number of pages | 14 |

Journal | Computational Geometry: Theory and Applications |

Volume | 69 |

DOIs | |

State | Published - 1 Jun 2018 |

## Keywords

- Discrete geometry
- Incidence geometry

## ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics