The number of unit-area triangles in the plane: Theme and variations

We show that the number of unit-area triangles determined by a set $S$ of $n$ points in the plane is $O(n^{20/9})$, improving the earlier bound $O(n^{9/4})$ of Apfelbaum and Sharir [Discrete Comput. Geom., 2010]. 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 $\Omega(n^2)$, for any triple of lines (it is always $O(n^2)$ in this case). (ii) We show that if $S$ is a {\em convex grid} of the form $A\times B$, where $A$, $B$ are {\em convex} sets of $n^{1/2}$ real numbers each (i.e., the sequences of differences of consecutive elements of $A$ and of $B$ are both strictly increasing), then $S$ determines $O(n^{31/14})$ unit-area triangles

Improved estimates on the number of unit perimeter triangles

We obtain new upper and lower bounds on the number of unit perimeter triangles spanned by points in the plane. We also establish improved bounds in the special case where the point set is a section of the integer grid.Comment: 8 pages, 1 figur

Kinetic and Dynamic Delaunay tetrahedralizations in three dimensions

We describe the implementation of algorithms to construct and maintain three-dimensional dynamic Delaunay triangulations with kinetic vertices using a three-simplex data structure. The code is capable of constructing the geometric dual, the Voronoi or Dirichlet tessellation. Initially, a given list of points is triangulated. Time evolution of the triangulation is not only governed by kinetic vertices but also by a changing number of vertices. We use three-dimensional simplex flip algorithms, a stochastic visibility walk algorithm for point location and in addition, we propose a new simple method of deleting vertices from an existing three-dimensional Delaunay triangulation while maintaining the Delaunay property. The dual Dirichlet tessellation can be used to solve differential equations on an irregular grid, to define partitions in cell tissue simulations, for collision detection etc.Comment: 29 pg (preprint), 12 figures, 1 table Title changed (mainly nomenclature), referee suggestions included, typos corrected, bibliography update

On distinct distances in homogeneous sets in the Euclidean space

A homogeneous set of $n$ points in the $d$-dimensional Euclidean space determines at least $\Omega(n^{2d/(d^2+1)} / \log^{c(d)} n)$ distinct distances for a constant $c(d)>0$. In three-space, we slightly improve our general bound and show that a homogeneous set of $n$ points determines at least $\Omega(n^{.6091})$ distinct distances

The Number of Unit-Area Triangles in the Plane: Theme and Variations *

Abstract We show that the number of unit-area triangles determined by a set S of n points in the plane is O(n 20/9 ), improving the earlier bound O(n 9/4 ) of Apfelbaum and Sharir (ii) We show that if S is a convex grid of the form A × B, where A, B are convex sets of n 1/2 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(n 31/14 ) unit-area triangles

Convex Polygons in Cartesian Products

We study several problems concerning convex polygons whose vertices lie in a Cartesian product of two sets of n real numbers (for short, grid). First, we prove that every such grid contains a convex polygon with Omega(log n) vertices and that this bound is tight up to a constant factor. We generalize this result to d dimensions (for a fixed d in N), and obtain a tight lower bound of Omega(log^{d-1}n) for the maximum number of points in convex position in a d-dimensional grid. Second, we present polynomial-time algorithms for computing the longest convex polygonal chain in a grid that contains no two points with the same x- or y-coordinate. We show that the maximum size of such a convex polygon can be efficiently approximated up to a factor of 2. Finally, we present exponential bounds on the maximum number of convex polygons in these grids, and for some restricted variants. These bounds are tight up to polynomial factors