Results

**1 - 3**of**3**### Collinearities in Kinetic Point Sets

, 2011

"... Let P be a set of n points in the plane, each point moving along a given trajectory. A k-collinearity is a pair (L, t) of a line L and a time t such that L contains at least k points at time t, L is spanned by the points at time t (i.e., the points along L are not all coincident), and not all of the ..."

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Let P be a set of n points in the plane, each point moving along a given trajectory. A k-collinearity is a pair (L, t) of a line L and a time t such that L contains at least k points at time t, L is spanned by the points at time t (i.e., the points along L are not all coincident), and not all of the points are collinear at all times. We show that, if the points move with constant velocity, then the number of 3-collinearities is at most 2 ()

### THE JOINTS PROBLEM IN n

, 906

"... n Abstract. We show that given a collection of A lines in, n � 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(An/(n−1)). An analogous result for smooth algebraic curves is also proven. 1. ..."

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n Abstract. We show that given a collection of A lines in, n � 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(An/(n−1)). An analogous result for smooth algebraic curves is also proven. 1.

### Incidence Theorems and Their Applications

"... We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: 1. Counting incidences: Given a set (or s ..."

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We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: 1. Counting incidences: Given a set (or several sets) of geometric objects (lines, points, etc.), what is the maximum number of incidences (or intersections) that can exist between elements in different sets? We will see several results of this type, such as the Szemeredi-Trotter theorem, over the reals and over finite fields and discuss their applications in combinatorics (e.g., in the recent solution of Guth and Katz to Erdos ’ distance problem) and in computer science (in explicit constructions of multi-source extractors). 2. Kakeya type problems: These problems deal with arrangements of lines that point in different directions. The goal is to try and understand to what extent these lines can overlap one another. We will discuss these questions both over the reals and over finite fields and see how they come up in the