Computing envelopes in four dimensions with applications

Pankaj K. Agarwal*, Boris Aronov, Micha Sharir

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


Let F be a collection of n d-variate, possibly partially defined, functions, all algebraic of some constant maximum degree. We present a randomized algorithm that computes the vertices, edges, and 2-faces of the lower envelope (i.e., pointwise minimum) of F in expected time O(nd+ε), for any ε > 0. For d = 3, by combining this algorithm with the point location technique of Preparata and Tamassia, we can compute, in randomized expected time O(n3+ε), for any ε > 0, a data structure of size O(n3+ε) that, given any query point q, can determine in O(log2 n) time whether q lies above, below or on the envelope. As a consequence, we obtain improved algorithmic solutions to many problems in computational geometry, including (a) computing the width of a point set in 3-space, (b) computing the biggest stick in a simple polygon in the plane, and (c) computing the smallest-width annulus covering a planar point set. The solutions to these problems run in time O(n17/11+ε), for any ε > 0, improving previous solutions that run in time O(n8/5+ε). We also present data structures for (i) performing nearest-neighbor and related queries for fairly general collections of objects in 3-space and for collections of moving objects in the plane, and (ii) performing ray-shooting and related queries among n spheres or more general objects in 3-space. Both of these data structures require O(n3+ε) storage and preprocessing time, for any ε > 0, and support polylogarithmic-time queries. These structures improve previous solutions to these problems.

Original languageEnglish
Title of host publicationProceedings of the Annual Symposium on Computational Geometry
PublisherAssociation for Computing Machinery (ACM)
Number of pages11
ISBN (Print)0897916484, 9780897916486
StatePublished - 1994
Externally publishedYes
EventProceedings of the 10th Annual Symposium on Computational Geometry - Stony Brook, NY, USA
Duration: 6 Jun 19948 Jun 1994

Publication series

NameProceedings of the Annual Symposium on Computational Geometry


ConferenceProceedings of the 10th Annual Symposium on Computational Geometry
CityStony Brook, NY, USA


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