On the Area Bisectors of a Polygon

We consider the family of lines that are area bisectors of a polygon (possibly with holes) in the plane. We say that two bisectors of a polygon P are combinatorially distinct if they induce di erent partitionings of the vertices of P . We derive an algebraic characterization of area bisectors. We then show that there are simple polygons with n vertices that have (n) combinatorially distinct area bisectors (matching the obvious upper bound), and present an output-sensitive algorithm for computing an explicit representation of all the bisectors of a given polygon. Work on this paper by Karl-Friedrich B ohringer and Bruce Randall Donald has been supported in part by the National Science Foundation under grants no. IRI-8802390, IRI-9000532, IRI-9201699, IRI-9530785, IRI-9896020, by a Presidential Young Investigator award to Bruce Donald, by an NSF/ARPA Small Grant for Exploratory Research no. IRI-9403903, by an NSF CISE Postdoctoral Associateship to Karl Bohringer no. CDA-9705022, and in part by the Air Force O ce of Sponsored Research, the Mathematical Sciences Institute, Intel Corporation, and AT&T Bell laboratories. Work on this paper by Dan Halperin has been supported in part by an Alon Fellowship, by ESPRIT IV LTR Project no. 21957 (CGAL), by the USA-Israel Binational Science Foundation, by The Israel Science Foundation founded by the Israel Academy of Sciences and Humanities, and by the Hermann Minkowski { Minerva Center for Geometry at Tel Aviv University. A preliminary and abridged version of the paper appeared in Proc. 13th ACM Symp. on Computational Geometry, Nice, 1997, pp. 457{459. ALPHA laboratory, Dept. of Ind. Eng. and Op. Research, University of California, Berkeley, CA 947201777, [email protected], www.ieor.berkeley.edu/~karl. Dept. of Computer Science, Dartmouth College, 6211 Sudiko Laboratory, Hanover, NH 03755-3510. [email protected], www.cs.dartmouth.edu/~brd. Department of Computer Science, Tel Aviv University, Tel Aviv 69978, ISRAEL, [email protected], www.math.tau.ac.il/~halperin. Part of the work on this paper was carried out while D.H. was at the Robotics Laboratory, Department of Computer Science, Stanford University.