Constructing strongly convex hulls using exact or rounded arithmetic

Research output: Contribution to journalArticle

15 Scopus citations

Abstract

One useful generalization of the convex hull of a set S of n points is the e{open}-strongly convex δ-hull. It is defined to be a convex polygon with vertices taken from S such that no point in S lies farther than δ outside and such that even if the vertices of are perturbed by as much as e{open}, remains convex. It was an open question as to whether an e{open}-strongly convex O(e{open})-hull existed for all positive e{open}. We give here an O(n log n) algorithm for constructing it (which thus proves its existence). This algorithm uses exact rational arithmetic. We also show how to construct an e{open}-strongly convex O(e{open} + μ)-hull in O(n log n) time using rounded arithmetic with rounding unit μ. This is the first rounded-arithmetic convex-hull algorithm which guarantees a convex output and which has error independent of n.

Original languageEnglish (US)
Pages (from-to)345-364
Number of pages20
JournalAlgorithmica
Volume8
Issue number1
DOIs
StatePublished - Jan 1 1992
Externally publishedYes

Keywords

  • Approximate arithmetic
  • Convex hulls
  • Exact rational arithmetic
  • Numerical analysis
  • Robust geometry
  • Strong and weak convexity

ASJC Scopus subject areas

  • Computer Science(all)
  • Computer Science Applications
  • Applied Mathematics

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