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 Affiliation 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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