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- W2959883145 abstract "We propose a geometric structure induced by any given convex polygon $P$, called $Nest(P)$, which is an arrangement of $Theta(n^2)$ line segments, each of which is parallel to an edge of $P$, where $n$ denotes the number of edges of $P$. We then deduce six nontrivial properties of $Nest(P)$ from the convexity of $P$ and the parallelism of the line segments in $Nest(P)$. Among others, we show that $Nest(P)$ is a subdivision of the exterior of $P$, and its inner boundary interleaves the boundary of $P$. They manifest that $Nest(P)$ has a surprisingly good interaction with the boundary of $P$. Furthermore, we study some computational problems on $Nest(P)$. In particular, we consider three kinds of location queries on $Nest(P)$ and answer each of them in (amortized) $O(log^2n)$ time. Our algorithm for answering these queries avoids an explicit construction of $Nest(P)$, which would take $Omega(n^2)$ time. By applying the aforementioned six properties altogether, we find that the geometric optimization problem of finding the maximum area parallelogram(s) in $P$ can be reduced to answering $O(n)$ aforementioned location queries, and thus be solved in $O(nlog^2n)$ time. This application will be reported in a subsequent paper." @default.
- W2959883145 created "2019-07-23" @default.
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- W2959883145 date "2015-12-12" @default.
- W2959883145 modified "2023-09-27" @default.
- W2959883145 title "A Geometric Structure Associated with the Convex Polygon" @default.
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