Related papers: Generalizations of Joints Problem

We study the "generic" degenerations of curves with two singular points when the points merge. First, the notion of generic degeneration is defined precisely. Then a method to classify the possible results of generic degenerations is…

We generalize the notions of flippable and simultaneously flippable edges in a triangulation of a set S of points in the plane to so-called \emph{pseudo-simultaneously flippable edges}. Such edges are related to the notion of convex…

The aim of this paper is to generalize Apollonius' problem. The problem is to construct a circle that is tangent to three given circles in a plane. We find the maximum possible number of solution circles in the case of more than the three…

We prove an incidence theorem for points and planes in the projective space $\mathbb P^3$ over any field $\mathbb F$, whose characteristic $p\neq 2.$ An incidence is viewed as an intersection along a line of a pair of two-planes from two…

We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set $S$ of $s$ points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in…

An ordinary plane of a finite set of points in real 3-space with no three collinear is a plane intersecting the set in exactly three points. We prove a structure theorem for sets of points spanning few ordinary planes. Our proof relies on…

Let P be a set of n points in the plane, not all on a line. We show that if n is large then there are at least n/2 ordinary lines, that is to say lines passing through exactly two points of P. This confirms, for large n, a conjecture of…

Multinets are certain configurations of lines and points with multiplicities in the complex projective plane P2. They are used in the studies of resonance and characteristic varieties of complex hyperplane arrangement complements and…

For an arrangement of $n$ lines in the real projective plane, we denote by $f$ the number of regions into which the real projective plane is divided by the lines. Using Bojanowski's inequality, we establish a new lower bound for $f$. In…

Let $P$ be a set of $n$ points in general position on the plane. A set of closed convex polygons with vertices in $P$, and with pairwise disjoint interiors is called a convex decomposition of $P$ if their union is the convex hull of $P$,…

We study the set of image tuples arising from fixed cameras observing varying planar 3-dimensional point configurations. We derive a formula for the number of complex critical points of the triangulation problem, which seeks to reconstruct…

We construct examples of nonresolvable generalized $n$-manifolds, $n\geq 6$, with arbitrary resolution obstruction, homotopy equivalent to any simply connected, closed $n$-manifold. We further investigate the structure of generalized…

We count the number of conics through two general points in complete intersections when this number is finite and give an application in terms of quasi-lines.

As a generalization of polyominoes we consider edge-to-edge connected nonoverlapping unions of regular $k$-gons. For $n\le 4$ we determine formulas for the number $a_k(n)$ of generalized polyominoes consisting of $n$ regular $k$-gons.…

We first review some topics in the classical computational geometry of lines, in particular the O(n^{3+\epsilon}) bounds for the combinatorial complexity of the set of lines in R^3 interacting with $n$ objects of fixed description…

We investigate combinatorial bounds for the total Tjurina numbers of plane curve arrangements. Focusing on arrangements of lines and conics in $\mathbb{P}^2$ that admit only ordinary quasi-homogeneous singularities, we derive new structural…

In this paper we study some Erdos type problems in discrete geometry. Our main result is that we show that there is a planar point set of n points such that no four are collinear but no matter how we choose a subset of size $n^{5/6+o(1)} $…

We propose and study a generalization to the well-known problem of polyline simplification. Instead of a single polyline, we are given a set of $\ell$ polylines possibly sharing some line segments and bend points. Our goal is to minimize…

If two Jordan curves in the plane have precisely one point in common, and there they do not properly cross, then the common point is called a {\em touching point}. The main result of this paper is a Crossing Lemma for simple curves: Let $X$…

We use three different methods to count the number of lines in the plane whose intersection with a fixed general quintic has fixed cross-ratios. We compare and contrast these methods, shedding light on some classical ideas which underly…