TY - JOUR
T1 - External polygon containment problems
AU - Sharir, Micha
AU - Toledo, Sivan
N1 - Funding Information:
*A preliminary version of the results in this paper appeared in [26]. Work on this paper by the first author has been supported by Office of Naval Research Grant NOO014-90-J-1284, by National Science Foundation Grant CCR-89-01484, and by grants from the U.S.-Israeli Binational Science Foundation, the G.I.F.-the German Israeli Foundation for Scientific Research and Development, and the Fund for Basic Research administered by the Israeli Academy of Sciences. *Corresponding author.
PY - 1994/6
Y1 - 1994/6
N2 - Given a convex polygonal object P with k vertices and an environment consisting of polygonal obstacles with a total of n corners, we seek a placement for the largest copy of P that does not intersect any of the obstacles, allowing translation, rotation and scaling. We employ the parametric search technique of Megiddo (1983), and the fixed size polygon placement algorithms developed by Leven and Sharir (1987), to obtain an algorithm that runs in time O(k2nλ6(kn)log3(kn)loglog(kn)). We also present several other efficient algorithms for restricted variants of the extremal polygon containment problem, using the same ideas. These variants include: placement of the largest homothetic copies of one or two convex polygons in another convex polygon and placement of the largest similar copy of a triangle in a convex polygon.
AB - Given a convex polygonal object P with k vertices and an environment consisting of polygonal obstacles with a total of n corners, we seek a placement for the largest copy of P that does not intersect any of the obstacles, allowing translation, rotation and scaling. We employ the parametric search technique of Megiddo (1983), and the fixed size polygon placement algorithms developed by Leven and Sharir (1987), to obtain an algorithm that runs in time O(k2nλ6(kn)log3(kn)loglog(kn)). We also present several other efficient algorithms for restricted variants of the extremal polygon containment problem, using the same ideas. These variants include: placement of the largest homothetic copies of one or two convex polygons in another convex polygon and placement of the largest similar copy of a triangle in a convex polygon.
KW - Motion planning
KW - Parametric searching
KW - Polygon containment
UR - http://www.scopus.com/inward/record.url?scp=0039855263&partnerID=8YFLogxK
U2 - 10.1016/0925-7721(94)90011-6
DO - 10.1016/0925-7721(94)90011-6
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AN - SCOPUS:0039855263
SN - 0925-7721
VL - 4
SP - 99
EP - 118
JO - Computational Geometry: Theory and Applications
JF - Computational Geometry: Theory and Applications
IS - 2
ER -