On the Distribution of Values of Angles Determined by Coplanar Points

Conway, James; Croft, H. T.; ERDijS, P.; Guy, M. J. T.

doi:10.1112/jlms/s2-19.1.137

Cited by 23 publications

(13 citation statements)

References 7 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In R 3 , the best known upper bound, O(n 8/3 ), is due to Conway et al [2]; see also [1,Section 6.2]. We improve this bound to O(n 7/3 ) and show that this bound is tight in case α = π/2.…”

mentioning

confidence: 72%

Repeated Angles in Three and Four Dimensions

Apfelbaum¹,

Sharir²

2005

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

mentioning

confidence: 72%

Repeated Angles in Three and Four Dimensions

Apfelbaum¹,

Sharir²

2005

SIAM J. Discrete Math.

View full text Add to dashboard Cite

show abstract

“…To prove the upper bound, we use a configuration from [12]. n points are arranged as equal subsets of n 3 points, each along small circular arcs at the vertices of a triangle ABC.…”

Section: General Point Setsmentioning

confidence: 99%

Selection Lemmas for Various Geometric Objects

Ashok

Govindarajan

Rajgopal

2016

Int. J. Comput. Geom. Appl.

View full text Add to dashboard Cite

Selection lemmas are classical results in discrete geometry that have been well studied and have applications in many geometric problems like weak epsilon nets and slimming Delaunay triangulations. Selection lemma type results typically show that there exists a point that is contained in many objects that are induced (spanned) by an underlying point set.In the first selection lemma, we consider the set of all the objects induced (spanned) by a point set P . This question has been widely explored for simplices in R d , with tight bounds in R 2 . In our paper, we prove first selection lemma for other classes of geometric objects. We also consider the strong variant of this problem where we add the constraint that the piercing point comes from P . We prove an exact result on the strong and the weak variant of the first selection lemma for axis-parallel rectangles, special subclasses of axis-parallel rectangles like quadrants and slabs, disks (for centrally symmetric point sets). We also show non-trivial bounds on the first selection lemma for axis-parallel boxes and hyperspheres in R d . In the second selection lemma, we consider an arbitrary m sized subset of the set of all objects induced by P . We study this problem for axisparallel rectangles and show that there exists an point in the plane that is contained in m 3 24n 4 rectangles. This is an improvement over the previous bound by Smorodinsky and Sharir [20] when m is almost quadratic.

show abstract

“…fd (< a) and fd ( >a) show at least what percent of the angles are smaller (or greater) than a for a pointset 91 in Ed . Convay, Croft, Erdős and Guy in [2] investigate f2 and f3 . They show, for instance, that…”

Section: P Elmentioning

confidence: 99%