Point-Location in The Arrangement of Curves

Curves in $\mathbb{R}^4$ and two-rich points

We obtain a new bound on the number of two-rich points spanned by an arrangement of low degree algebraic curves in $\mathbb{R}^4$. Specifically, we show that an arrangement of $n$ algebraic curves determines at most $C_εn^{4/3+3ε}$ two-rich points, provided at most $n^{2/3+2ε}$ curves lie in any low degree hypersurface and at most $n^{1/3+ε}$ curves lie in any low degree surface. This result follows from a structure theorem about arrangements of curves that determine many two-rich points.

New bounds on curve tangencies and orthogonalities

We establish new bounds on the number of tangencies and orthogonal intersections determined by an arrangement of curves. First, given a set of $n$ algebraic plane curves, we show that there are $O(n^{3/2})$ points where two or more curves are tangent. In particular, if no three curves are mutually tangent at a common point, then there are $O(n^{3/2})$ curve-curve tangencies. Second, given a family of algebraic plane curves and a set of $n$ curves from this family, we show that either there are $O(n^{3/2})$ points where two or more curves are orthogonal, or the family of curves has certain spec…

Lower bounds for Arrangement-based Range-Free Localization in Sensor Networks

Colander are location aware entities that collaborate to determine approximate location of mobile or static objects when beacons from an object are received by all colanders that are within its distance $R$. This model, referred to as arrangement-based localization, does not require distance estimation between entities, which has been shown to be highly erroneous in practice. Colander are applicable in localization in sensor networks and tracking of mobile objects. A set $S \subset {\mathbb R}^2$ is an $(R,ε)$-colander if by placing receivers at the points of $S$, a wireless device with transm…

Planar $β$-skeletons via point location in monotone subdivisions of subset of lunes

We present a new algorithm for lune-based $β$-skeletons for sets of $n$ points in the plane, for $β\in (2,\infty]$, the only case when optimal algorithms are not known. The running time of the algorithm is $O(n^{3/2} \log^{1/2} n)$, which is the best known and is an improvement of Rao and Mukhopadhyay \cite{rm97} result. The method is based on point location in monotonic subdivisions of arrangements of curve segments.

Cutting Algebraic Curves into Pseudo-segments and Applications

We show that a set of $n$ algebraic plane curves of constant maximum degree can be cut into $O(n^{3/2}\operatorname{polylog} n)$ Jordan arcs, so that each pair of arcs intersect at most once, i.e., they form a collection of pseudo-segments. This extends a similar (and slightly better) bound for pseudo-circles due to Marcus and Tardos. Our result is based on a technique of Ellenberg, Solymosi and Zahl that transforms arrangements of plane curves into arrangements of space curves, so that lenses (pairs of subarcs of the curves that intersect at least twice) become vertical depth cycles. We then …

Arrangement Computation for Planar Algebraic Curves

We present a new certified and complete algorithm to compute arrangements of real planar algebraic curves. Our algorithm provides a geometric-topological analysis of the decomposition of the plane induced by a finite number of algebraic curves in terms of a cylindrical algebraic decomposition of the plane. Compared to previous approaches, we improve in two main aspects: Firstly, we significantly reduce the amount of exact operations, that is, our algorithms only uses resultant and gcd as purely symbolic operations. Secondly, we introduce a new hybrid method in the lifting step of our algorithm…

Arrangements of Pseudocircles: On Circularizability

An arrangement of pseudocircles is a collection of simple closed curves on the sphere or in the plane such that any two of the curves are either disjoint or intersect in exactly two crossing points. We call an arrangement intersecting if every pair of pseudocircles intersects twice. An arrangement is circularizable if there is a combinatorially equivalent arrangement of circles. In this paper we present the results of the first thorough study of circularizability. We show that there are exactly four non-circularizable arrangements of 5 pseudocircles (one of them was known before). In the set o…

伪直线和伪圆的$k$-排列

$k$-arrangements of pseudolines and pseudocircles

本文证明对每个$m\ge 3$，大小为$m$的不可避免的简单$k$-伪直线排列数量随$k$指数增长，且对偶数$k$，伪圆的$k$-排列也有类似结果。

扫描 $x$-单调伪直线

Sweeping $x$-monotone pseudolines

针对 $x$-单调伪直线排列的绳扫描问题，给出绳长上界 $2n-2$ 与下界 $7(n-2)/4+1$，并探讨了最短绳长扫描的复杂度相关问题。

关于求解简单曲线数独

On Solving Simple Curved Nonograms

本文针对曲线数独的求解问题，填补了以往研究仅关注其创建的空白，为不同复杂度的曲线数独提供了算法和硬度结果。

Distinct Distances on Curves via Rigidity

It is shown that $N$ points on a real algebraic curve of degree $n$ in $\mathbb{R}^d$ always determine $\gtrsim_{n,d}N^{1+\frac{1}{4}}$ distinct distances, unless the curve is a straight line or the closed geodesic of a flat torus. In the latter case, there are arrangements of $N$ points which determine $\lesssim N$ distinct distances. The method may be applied to other quantities of interest to obtain analogous exponent gaps. An important step in the proof involves understanding the structural rigidity of certain frameworks on curves.