TY - JOUR

T1 - Rotational polygon containment and minimum enclosure using only robust 2D constructions

AU - Milenkovic, Victor J.

N1 - Funding Information:
I Expanded version of a paper presented at the 14th ACM Symposium on Computational Geometry, June 1998. ∗E-mail: vjm@cs.miami.edu; http://www.cs.miami.edu 1This research was funded by the Textile/Clothing Technology Corporation from funds awarded by the Alfred P. Sloan Foundation, by NSF grants CCR-91-157993 and NSF-CCR-97-12401, and by a subcontract of a National Textile Center grant to Auburn University, Department of Consumer Affairs.

PY - 1999/5

Y1 - 1999/5

N2 - 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.

AB - 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.

KW - Containment

KW - Geometric rounding

KW - Layout

KW - Minimum enclosure

KW - Packing or nesting of irregular polygons

KW - Robust geometry

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U2 - 10.1016/S0925-7721(99)00006-1

DO - 10.1016/S0925-7721(99)00006-1

M3 - Article

AN - SCOPUS:0001551468

VL - 13

SP - 3

EP - 19

JO - Computational Geometry: Theory and Applications

JF - Computational Geometry: Theory and Applications

SN - 0925-7721

IS - 1

ER -