article published yes Herbert Edelsbrunner author 3FB178DA-F248-11E8-B48F-1D18A9856A870000-0002-9823-6833 Joseph O'Rourke author Raimund Seidel author 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 SIAM1986 eng 0097-5397 1095-711110.1137/0215024 152341 - 363 yes Edelsbrunner, Herbert, et al. “Constructing Arrangements of Lines and Hyperplanes with Applications.” <i>SIAM Journal on Computing</i>, vol. 15, no. 2, SIAM, 1986, pp. 341–63, doi:<a href="https://doi.org/10.1137/0215024">10.1137/0215024</a>. Edelsbrunner, Herbert, Joseph O’Rourke, and Raimund Seidel. “Constructing Arrangements of Lines and Hyperplanes with Applications.” <i>SIAM Journal on Computing</i>. SIAM, 1986. <a href="https://doi.org/10.1137/0215024">https://doi.org/10.1137/0215024</a>. Edelsbrunner H, O’Rourke J, Seidel R. 1986. Constructing arrangements of lines and hyperplanes with applications. SIAM Journal on Computing. 15(2), 341–363. Edelsbrunner H, O’Rourke J, Seidel R. Constructing arrangements of lines and hyperplanes with applications. <i>SIAM Journal on Computing</i>. 1986;15(2):341-363. doi:<a href="https://doi.org/10.1137/0215024">10.1137/0215024</a> H. Edelsbrunner, J. O’Rourke, and R. Seidel, “Constructing arrangements of lines and hyperplanes with applications,” <i>SIAM Journal on Computing</i>, vol. 15, no. 2. SIAM, pp. 341–363, 1986. Edelsbrunner, H., O’Rourke, J., &#38; Seidel, R. (1986). Constructing arrangements of lines and hyperplanes with applications. <i>SIAM Journal on Computing</i>. SIAM. <a href="https://doi.org/10.1137/0215024">https://doi.org/10.1137/0215024</a> H. Edelsbrunner, J. O’Rourke, R. Seidel, SIAM Journal on Computing 15 (1986) 341–363. 41052018-12-11T12:06:58Z2022-02-01T11:03:07Z