TY - JOUR

T1 - Distinct distances between points and lines

AU - Sharir, Micha

AU - Smorodinsky, Shakhar

AU - Valculescu, Claudiu

AU - de Zeeuw, Frank

N1 - Publisher Copyright:
© 2017 Elsevier B.V.

PY - 2018/6

Y1 - 2018/6

N2 - 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 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(m4/3). 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(n3/2) points where two or more circles are tangent, improving the previously best known bound of O(n3/2logn). 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.

AB - 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 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(m4/3). 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(n3/2) points where two or more circles are tangent, improving the previously best known bound of O(n3/2logn). 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.

KW - Discrete geometry

KW - Incidence geometry

UR - http://www.scopus.com/inward/record.url?scp=85034061712&partnerID=8YFLogxK

U2 - 10.1016/j.comgeo.2017.10.008

DO - 10.1016/j.comgeo.2017.10.008

M3 - מאמר

AN - SCOPUS:85034061712

VL - 69

SP - 2

EP - 15

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

ER -