Rotational polygon containment and minimum enclosure using only robust 2D constructions

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40 Citations (Scopus)

Abstract

An algorithm and a robust floating point implementation is given for rotational polygon containment: given polygons P1, P2, P3, . . ., Pk and a container polygon C, find rotations and translations for the k polygons that place them into the container without overlapping. A version of the algorithm and implementation also solves rotational minimum enclosure: given a class C of container polygons, find a container C ∈ C of minimum area for which containment has a solution. The minimum enclosure is approximate: it bounds the minimum area between (1 - ε)A and A. Experiments indicate that finding the minimum enclosure is practical for k = 2, 3 but not larger unless optimality is sacrificed or angles ranges are limited (although these solutions can still be useful). Important applications for these algorithm to industrial problems are discussed. The paper also gives practical algorithms and numerical techniques for robustly calculating polygon set intersection, Minkowski sum, and range intersection: the intersection of a polygon with itself as it rotates through a range of angles. In particular, it introduces nearest pair rounding, which allows all these calculations to be carried out in rounded floating point arithmetic.

Original languageEnglish (US)
Pages (from-to)3-19
Number of pages17
JournalComputational Geometry: Theory and Applications
Volume13
Issue number1
StatePublished - May 1999

Fingerprint

Enclosure
Enclosures
Polygon
Containers
Container
Digital arithmetic
Intersection
Range of data
Minkowski Sum
Angle
Floating-point Arithmetic
Rounding
Floating point
Numerical Techniques
Overlapping
Optimality
Experiments
Experiment

Keywords

  • Containment
  • Geometric rounding
  • Layout
  • Minimum enclosure
  • Packing or nesting of irregular polygons
  • Robust geometry

ASJC Scopus subject areas

  • Computational Theory and Mathematics
  • Discrete Mathematics and Combinatorics
  • Geometry and Topology

Cite this

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abstract = "An algorithm and a robust floating point implementation is given for rotational polygon containment: given polygons P1, P2, P3, . . ., Pk and a container polygon C, find rotations and translations for the k polygons that place them into the container without overlapping. A version of the algorithm and implementation also solves rotational minimum enclosure: given a class C of container polygons, find a container C ∈ C of minimum area for which containment has a solution. The minimum enclosure is approximate: it bounds the minimum area between (1 - ε)A and A. Experiments indicate that finding the minimum enclosure is practical for k = 2, 3 but not larger unless optimality is sacrificed or angles ranges are limited (although these solutions can still be useful). Important applications for these algorithm to industrial problems are discussed. The paper also gives practical algorithms and numerical techniques for robustly calculating polygon set intersection, Minkowski sum, and range intersection: the intersection of a polygon with itself as it rotates through a range of angles. In particular, it introduces nearest pair rounding, which allows all these calculations to be carried out in rounded floating point arithmetic.",
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