On the Area Bisectors of a Polygon

Abstract. 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 combinatoriallydistinct if they induce different 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 Ω(n2) 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.

[1]  Gregory T. A. Kovacs,et al.  Computational methods for design and control of MEMS micromanipulator arrays , 1997 .

[2]  J. O´Rourke,et al.  Computational Geometry in C: Arrangements , 1998 .

[3]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[4]  Thomas C. Shermer.  A Linear Algorithm for Bisecting a Polygon , 1992, Inf. Process. Lett..

[5]  Leonidas J. Guibas,et al.  Arrangements of Curves in the Plane - Topology, Combinatorics and Algorithms , 2018, Theor. Comput. Sci..

[6]  Gert Vegter,et al.  In handbook of discrete and computational geometry , 1997 .

[7]  Ken Goldberg,et al.  Sensorless Manipulation Using Transverse Vibrations of a Plate , 1995, Proceedings of 1995 IEEE International Conference on Robotics and Automation.

[8]  Jan van Leeuwen,et al.  Maintenance of Configurations in the Plane , 1981, J. Comput. Syst. Sci..

[9]  Karl-Friedrich Bohringer,et al.  Programmable force fields for distributed manipulation, and their implementation using micro-fabricated actuator arrays , 1997 .

[10]  F. Frances Yao,et al.  Computational Geometry , 1991, Handbook of Theoretical Computer Science, Volume A: Algorithms and Complexity.

[11]  Tamal K. Dey,et al.  Improved bounds on planar k-sets and k-levels , 1997, Proceedings 38th Annual Symposium on Foundations of Computer Science.

[12]  Herbert Edelsbrunner,et al.  Algorithms in Combinatorial Geometry , 1987, EATCS Monographs in Theoretical Computer Science.

[13]  Herbert Edelsbrunner,et al.  On the Number of Line Separations of a Finite Set in the Plane , 1985, J. Comb. Theory, Ser. A.

[14]  Leonidas J. Guibas,et al.  Primitives for the manipulation of general subdivisions and the computation of Voronoi diagrams , 1983, STOC.

[15]  N. C. MacDonald,et al.  Upper and Lower Bounds for Programmable Vector Fields with Applications to MEMS and Vibratory Plate Parts Feeders , 1996 .

[16]  Joseph O'Rourke,et al.  Computational Geometry in C. , 1995 .

[17]  Matthew Galen Diaz.  Algorithms for balanced partitioning of polygons and point sets , 1991 .