Constructing arrangements of lines and hyperplanes with applications Journal Article


Author(s): Edelsbrunner, Herbert; O'Rourke, Joseph; Seidel, Raimund
Article Title: Constructing arrangements of lines and hyperplanes with applications
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Abstract: A finite set of lines partitions the Euclidean plane into a cell complex. Similarly, a finite set of $(d - 1)$-dimensional hyperplanes partitions $d$-dimensional Euclidean space. An algorithm is presented that constructs a representation for the cell complex defined by $n$ hyperplanes in optimal $O(n^d )$ time in $d$ dimensions. It relies on a combinatorial result that is of interest in its own right. The algorithm is shown to lead to new methods for computing $\lambda $-matrices, constructing all higher-order Voronoi diagrams, halfspatial range estimation, degeneracy testing, and finding minimum measure simplices. In all five applications, the new algorithms are asymptotically faster than previous results, and in several cases are the only known methods that generalize to arbitrary dimensions. The algorithm also implies an upper bound of $2^{cn^d } $, $c$ a positive constant, for the number of combinatorially distinct arrangements of $n$ hyperplanes in $E^d $. © 1986 Society for Industrial and Applied Mathematics
Keywords: computational geometry; arrangements; configurations; geometric transformation; combinatorial geometry; optimal algorithm
Journal Title: SIAM Journal on Computing
Volume: 15
Issue 2
ISSN: 0097-5397
Publisher: SIAM  
Date Published: 1986-01-01
Start Page: 341
End Page: 363
DOI: 10.1137/0215024
Open access: no
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