Voronoi Diagram of Polygonal Chains under the Discrete Fréchet Distance

Details Voronoi Diagram of Polygonal Chains under the Discrete Fréchet Distance Voronoi Diagram of Polygonal Chains under the Discrete Fréchet Distance

2007-05-19

arxiv.org/abs/0705.2835v1

Computer Science

Computational Geometry

13 pages, 2 figures

Scientific paper

Polygonal chains are fundamental objects in many applications like pattern recognition and protein structure alignment. A well-known measure to characterize the similarity of two polygonal chains is the famous Fr\`{e}chet distance. In this paper, for the first time, we consider the Voronoi diagram of polygonal chains in $d$-dimension ($d=2,3$) under the discrete Fr\`{e}chet distance. Given $n$ polygonal chains ${\cal C}$ in $d$-dimension ($d=2,3$), each with at most $k$ vertices, we prove fundamental properties of such a Voronoi diagram {\em VD}$_F({\cal C})$ by presenting the first known upper and lower bounds for {\em VD}$_F({\cal C})$.

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