TY - JOUR
T1 - Distinct distances between points and lines
AU - Sharir, Micha
AU - Smorodinsky, Shakhar
AU - Valculescu, Claudiu
AU - de Zeeuw, Frank
N1 - Funding Information:
Part of this research was performed while the first, second, and fourth authors were visiting the Institute for Pure and Applied Mathematics (IPAM) in Los Angeles, which is supported by the National Science Foundation .
Publisher Copyright:
© 2017 Elsevier B.V.
PY - 2018/6/1
Y1 - 2018/6/1
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 - Article
AN - SCOPUS:85034061712
SN - 0925-7721
VL - 69
SP - 2
EP - 15
JO - Computational Geometry: Theory and Applications
JF - Computational Geometry: Theory and Applications
ER -