New bounds for lower envelopes in three dimensions, with applications to visibility in terrains

Dan Halperin*, Micha Sharir

*Corresponding author for this work

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

7 Scopus citations

Abstract

We consider the problem of bounding the complexity of the lower envelope of n surface patches in 3-space, all algebraic of constant maximum degree, and bounded by algebraic arcs of constant maximum degree, with the additional property that the interiors of any triple of these surfaces intersect in at most two points. We show that the number of vertices on the lower envelope of n such surface patches is O(n2 · 2clog n), for some constant c depending on the shape and degree of the surface patches. We apply this result to obtain an upper bound on the combinatorial complexity of the 'lower envelope' of the space of all rays in 3-space that lie above a given polyhedral terrain K with n edges. This envelope consists of all rays that touch the terrain (but otherwise lie above it). We show that the combinatorial complexity of this ray-envelope is O(n3 · 2clog n) for some constant c; in particular, there are at most that many rays that pass above the terrain and touch it in 4 edges. This bound, combined with the analysis of de Berg et al. [2], gives an upper bound (which is almost tight in the worst case) on the number of topologically-different orthographic views of such a terrain.

Original languageEnglish
Title of host publicationProceedings of the 9th Annual Symposium on Computational Geometry
PublisherAssociation for Computing Machinery (ACM)
Pages11-18
Number of pages8
ISBN (Print)0897915828, 9780897915823
DOIs
StatePublished - 1993
Externally publishedYes
EventProceedings of the 9th Annual Symposium on Computational Geometry - San Diego, CA, USA
Duration: 19 May 199321 May 1993

Publication series

NameProceedings of the 9th Annual Symposium on Computational Geometry

Conference

ConferenceProceedings of the 9th Annual Symposium on Computational Geometry
CitySan Diego, CA, USA
Period19/05/9321/05/93

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