论文标题
在确定形状的布置中，关于线和斜率的一些开放问题
Some Open Problems Regarding the Number of Lines and Slopes in Arrangements that Determine Shapes
论文作者
论文摘要
欧几里得飞机上的一套直线和一套$ p $点点定义安排$ \ nathcal {a} $ =（$ l $，$ p $）的建筑线和注册标记，并且仅当$ p $中的任何点是$ p $中的任何点时，至少是$ l $ lonnon n nontare and n nontare and Bare and Bare and Bare and Bare and Bare and Bare and Bare and Parter的一点点（2） $ P $。这篇说明性文章讨论了有关这种点线安排的以下开放问题。假设$ k \ geq 0 $在飞机中给出了点数。必须确定多少个建筑线$ K $点？ $ k $点确定的构造线定义了多少个不同的斜率或方向？飞机的分区有多少个不同的施工线，以便以$ k $点相交？据报道，少量$ K $的经验证据为这三个问题提供了部分答案。在检查了有关发射率几何形状的有限线性空间的相关问题之后，在构造线的数量上也陈述了第一个问题的猜想。本文有助于与形状语法理论领域形状数学有关的工作体系。
A set $L$ of straight lines and a set $P$ of points in the Euclidean plane define an arrangement $\mathcal{A}$ = ($L$, $P$) of construction lines and registration marks, if and only if: (1) any point in $P$ is a point of intersection of at least two lines in $L$, and (2) any two nonparallel lines in $L$ have a unique point of intersection in $P$. This expository article discusses the following open problems regarding such point-line arrangements. Suppose $k \geq 0$ number of points are given in the plane. How many construction lines $k$ points must determine? How many distinct slopes, or directions, are defined by construction lines that $k$ points determine? How many distinct sets of construction lines partition the plane, such that the lines meet at exactly $k$ points? Empirical evidence is reported for small numbers of $k$, offering partial answers to the three problems. A conjecture is also stated for the first problem, on the number of construction lines, after examining a related problem about finite linear spaces from incidence geometry. This paper contributes to the body of work related to the mathematics of shapes in the area of shape grammar theory.
