TY - GEN

T1 - The Number of Unit-Area Triangles in the Plane

T2 - 31st International Symposium on Computational Geometry, SoCG 2015

AU - Raz, Orit E.

AU - Sharir, Micha

PY - 2015/6/1

Y1 - 2015/6/1

N2 - We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n20/9), improving the earlier bound O(n9/4) of Apfelbaum and Sharir [2]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if S consists of points on three lines, the number of unit-area triangles that S spans can be ω(n2), for any triple of lines (it is always O(n2) in this case). (ii) We show that if S is a convex grid of the form A × B, where A, B are convex sets of n1/2 real numbers each (i.e., the sequences of differences of consecutive elements of A and of B are both strictly increasing), then S determines O(n31/14) unit-area triangles.

AB - We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n20/9), improving the earlier bound O(n9/4) of Apfelbaum and Sharir [2]. We also consider two special cases of this problem: (i) We show, using a somewhat subtle construction, that if S consists of points on three lines, the number of unit-area triangles that S spans can be ω(n2), for any triple of lines (it is always O(n2) in this case). (ii) We show that if S is a convex grid of the form A × B, where A, B are convex sets of n1/2 real numbers each (i.e., the sequences of differences of consecutive elements of A and of B are both strictly increasing), then S determines O(n31/14) unit-area triangles.

KW - Combinatorial geometry

KW - Incidences

KW - Repeated configurations

UR - http://www.scopus.com/inward/record.url?scp=84958167209&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.SOCG.2015.569

DO - 10.4230/LIPIcs.SOCG.2015.569

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AN - SCOPUS:84958167209

T3 - Leibniz International Proceedings in Informatics, LIPIcs

SP - 569

EP - 583

BT - 31st International Symposium on Computational Geometry, SoCG 2015

A2 - Pach, Janos

A2 - Pach, Janos

A2 - Arge, Lars

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

Y2 - 22 June 2015 through 25 June 2015

ER -