## Abstract

The collinearity testing problem is a basic problem in computational geometry, in which, given three sets A, B, C in the plane, of n points each, the task is to detect a collinear triple of points in A×B×C or report there is no such triple. In this paper we consider a preprocessing variant of this question, namely, the collinearity indexing problem, in which we are given two sets A and B, each of n points in the plane, and our goal is to preprocess A and B into a data structure, so that, for any query point q∈R^{2}, we can determine whether q is collinear with a pair of points (a,b)∈A×B. We provide a solution to the problem for the case where the points of A, B lie on an integer grid, and the query points lie on a vertical line, with a data structure of subquadratic storage and sublinear query time. We then extend our result to the case where the query points lie on the graph of a polynomial of constant degree. Our solution is based on the function-inversion technique of Fiat and Naor [11].

Original language | English (US) |
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Article number | 101963 |

Journal | Computational Geometry: Theory and Applications |

Volume | 110 |

DOIs | |

State | Published - Mar 2023 |

## Keywords

- 3SUM-indexing
- Collinearity testing
- Computational geometry
- Function inversion

## ASJC Scopus subject areas

- Computer Science Applications
- Geometry and Topology
- Control and Optimization
- Computational Theory and Mathematics
- Computational Mathematics