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Published in:
2013 | OriginalPaper | Chapter
The Distance 4-Sector of Two Points Is Unique
Authors : Robert Fraser, Meng He, Akitoshi Kawamura, Alejandro López-Ortiz, J. Ian Munro, Patrick K. Nicholson
Published in: Algorithms and Computation
Publisher: Springer Berlin Heidelberg
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The (distance)
k
-sector is a generalization of the concept of bisectors proposed by Asano, Matoušek and Tokuyama. We prove the uniqueness of the 4-sector of two points in the Euclidean plane. Despite the simplicity of the unique 4-sector (which consists of a line and two parabolas), our proof is quite non-trivial. We begin by establishing uniqueness in a small region of the plane, which we show may be perpetually expanded afterward.