### Abstract

Let S be a connected planar polygonal subdivision with n edges that we want to preprocess for point-location queries, and where we are given the probability γ_{i} that the query point lies in a polygon P_{i} of S. We show how to preprocess S such that the query time for a point p∈P_{i} depends on γ_{i} and, in addition, on the distance from p to the boundary of P_{i} - the further away from the boundary, the faster the query. More precisely, we show that a point-location query can be answered in time O(min (logn, 1+log area(P_{i})/γ_{i}Δ_{p}^{2})), where Δ_{p} is the shortest Euclidean distance of the query point p to the boundary of P_{i}. Our structure uses O(n) space and O(nlogn) preprocessing time. It is based on a decomposition of the regions of S into convex quadrilaterals and triangles with the following property: for any point p ∈ P_{i}, the quadrilateral or triangle containing p has area Ω(Δ_{p}^{2}). For the special case where S is a subdivision of the unit square and γ_{i}=area(P_{i}), we present a simpler solution that achieves a query time of O(min(logn,log 1/Δ_{p}^{2})). The latter solution can be extended to convex subdivisions in three dimensions.

Original language | English (US) |
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Pages (from-to) | 17-31 |

Number of pages | 15 |

Journal | Computational Geometry: Theory and Applications |

Volume | 54 |

DOIs | |

State | Published - Apr 1 2016 |

### Keywords

- Mesh generation
- Point location
- Quadtree

### ASJC Scopus subject areas

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

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## Cite this

*Computational Geometry: Theory and Applications*,

*54*, 17-31. https://doi.org/10.1016/j.comgeo.2016.02.001