Searching for Point Locations using Lines

TL;DR

This paper addresses the problem of locating n unknown points on a plane using lines with specific slopes, establishing that n+1 slopes are necessary and providing an algorithm for the task.

Contribution

It proves that n+1 slopes are sufficient and necessary to determine all point locations and introduces an algorithm to achieve this.

Findings

Minimum slopes needed to locate points is n+1

Provided an algorithm for point location using slopes

Established theoretical bounds for the problem

Abstract

Versions of the following problem appear in several topics such as Gamma Knife radiosurgery, studying objects with the X-ray transform, the 3SUM problem, and the $k$-linear degeneracy testing. Suppose there are $n$ points on a plane whose specific locations are unknown. We are given all the lines that go through the points with a given slope. We show that the minimum number of slopes needed, in general, to find all the point locations is $n+1$ and we provide an algorithm to do so.