---
res:
bibo_abstract:
- 'We consider several problems involving points and planes in three dimensions.
Our main results are: (i) The maximum number of faces boundingm distinct cells
in an arrangement ofn planes isO(m 2/3 n logn +n 2); we can calculatem such cells
specified by a point in each, in worst-case timeO(m 2/3 n log3 n+n 2 logn). (ii)
The maximum number of incidences betweenn planes andm vertices of their arrangement
isO(m 2/3 n logn+n 2), but this number is onlyO(m 3/5– n 4/5+2 +m+n logm), for
any>0, for any collection of points no three of which are collinear. (iii)
For an arbitrary collection ofm points, we can calculate the number of incidences
between them andn planes by a randomized algorithm whose expected time complexity
isO((m 3/4– n 3/4+3 +m) log2 n+n logn logm) for any>0. (iv) Givenm points andn
planes, we can find the plane lying immediately below each point in randomized
expected timeO([m 3/4– n 3/4+3 +m] log2 n+n logn logm) for any>0. (v) The maximum
number of facets (i.e., (d–1)-dimensional faces) boundingm distinct cells in an
arrangement ofn hyperplanes ind dimensions,d>3, isO(m 2/3 n d/3 logn+n d–1).
This is also an upper bound for the number of incidences betweenn hyperplanes
ind dimensions andm vertices of their arrangement. The combinatorial bounds in
(i) and (v) and the general bound in (ii) are almost tight.@eng'
bibo_authorlist:
- foaf_Person:
foaf_givenName: Herbert
foaf_name: Edelsbrunner, Herbert
foaf_surname: Edelsbrunner
foaf_workInfoHomepage: http://www.librecat.org/personId=3FB178DA-F248-11E8-B48F-1D18A9856A87
orcid: 0000-0002-9823-6833
- foaf_Person:
foaf_givenName: Leonidas
foaf_name: Guibas, Leonidas
foaf_surname: Guibas
- foaf_Person:
foaf_givenName: Micha
foaf_name: Sharir, Micha
foaf_surname: Sharir
bibo_doi: 10.1007/BF02187785
bibo_issue: '1'
bibo_volume: 5
dct_date: 1990^xs_gYear
dct_isPartOf:
- http://id.crossref.org/issn/0179-5376
- http://id.crossref.org/issn/1432-0444
dct_language: eng
dct_publisher: Springer@
dct_title: The complexity of many cells in arrangements of planes and related problems@
...