TY - JOUR

T1 - On arrangements of Jordan arcs with three intersections per pair

AU - Edelsbrunner, Herbert

AU - Guibas, Leonidas

AU - Hershberger, John

AU - Pach, Janos

AU - Pollack, Richard

AU - Seidel, Raimund

AU - Sharir, Micha

AU - Snoeyink, Jack

PY - 1989/12

Y1 - 1989/12

N2 - Motivated by a number of motion-planning questions, we investigate in this paper some general topological and combinatorial properties of the boundary of the union of n regions bounded by Jordan curves in the plane. We show that, under some fairly weak conditions, a simply connected surface can be constructed that exactly covers this union and whose boundary has combinatorial complexity that is nearly linear, even though the covered region can have quadratic complexity. In the case where our regions are delimited by Jordan acrs in the upper halfplane starting and ending on the x-axis such that any pair of arcs intersect in at most three points, we prove that the total number of subarcs that appear on the boundary of the union is only Θ(nα(n)), where α(n) is the extremely slowly growing functional inverse of Ackermann's function.

AB - Motivated by a number of motion-planning questions, we investigate in this paper some general topological and combinatorial properties of the boundary of the union of n regions bounded by Jordan curves in the plane. We show that, under some fairly weak conditions, a simply connected surface can be constructed that exactly covers this union and whose boundary has combinatorial complexity that is nearly linear, even though the covered region can have quadratic complexity. In the case where our regions are delimited by Jordan acrs in the upper halfplane starting and ending on the x-axis such that any pair of arcs intersect in at most three points, we prove that the total number of subarcs that appear on the boundary of the union is only Θ(nα(n)), where α(n) is the extremely slowly growing functional inverse of Ackermann's function.

UR - http://www.scopus.com/inward/record.url?scp=0012681942&partnerID=8YFLogxK

U2 - 10.1007/BF02187745

DO - 10.1007/BF02187745

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AN - SCOPUS:0012681942

SN - 0179-5376

VL - 4

SP - 523

EP - 539

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

IS - 1

ER -