On arrangements of Jordan arcs with three intersections per pair

Herbert Edelsbrunner*, Leonidas Guibas, John Hershberger, Janos Pach, Richard Pollack, Raimund Seidel, Micha Sharir, Jack Snoeyink

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)523-539
Number of pages17
JournalDiscrete and Computational Geometry
Issue number1
StatePublished - Dec 1989


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