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Weisheng Si 2015-03-02 13:12:41 +11:00
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Cone_spanners_2/doc/Cone_spanners_2/Cone_spanners_2.txt
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@ -9,11 +9,11 @@ namespace CGAL {
This chapter describes classes for constructing two kinds of cone-based spanners (Yao graph and Theta
graph) given a set of vertices on the plane. Both exact and inexact constructions can be performed.
In exact construction, the cone boundaries are calculated using roots of polynomials (requiring `CORE::Expr` or `LEDA::real`). In inexact construction, the cone boundaries are calculated using an approximate Pi (3.14159265358979323846), which is still accurate enough for most applications. Moreover, this chapter describes functions for generating the data and script files used by Gnuplot to plot the constructed graphs.
In exact construction, the cone boundaries are calculated using roots of polynomials (requiring `CORE::Expr` or `LEDA::real`). In inexact construction, the cone boundaries are calculated using an approximate \f$ \pi \f$ (3.14159265358979323846), which is still accurate enough for most applications. Moreover, this chapter describes functions for generating the data and script files used by Gnuplot to plot the constructed graphs.
\section sec_CBS_mydefinitions Definitions
Given a set \f$V\f$ of vertices on the plane, the directed Yao Graph with an integer parameter \f$k (k > 1)\f$ on \f$V\f$ is obtained as follows. For each vertex \f$u \in V\f$ starting from a given direction (e.g., the direction of positive \f$y\f$-axis), draw \f$k\f$ equally-spaced rays \f$l_0\f$, \f$l_1\f$, \ldots, \f$l_{k-1}\f$ originating from \f$u\f$ in clockwise order (see \cgalFigureRef{f-y5} (a)). These rays divide the plane into \f$k\f$ cones of angle \f$2\pi/k\f$, denoted by \(c(u, 1), c(u, 2), \ldots, c(u, k)\) respectively in clockwise order. To avoid overlapping at boundaries, it is assumed here that the area of \(c(u, i)\), where \(i=1, \ldots, k\), includes the ray \f$l_{i-1}\f$ but includes the ray \f$l_{i\% k}\f$. In each cone of \f$u\f$, draw a directed edge from \f$u\f$ to its closest vertex by Euclidean distance in that cone. Ties are broken arbitrarily. These directed edges will form the edge set of the directed Yao graph on \f$V\f$. The undirected Yao Graph on \f$V\f$ is obtained by ignoring the directions of the edges. Note that if both edge \f$uv\f$ and \f$vu\f$ are in the directed Yao graph, only one edge \f$uv\f$ exists in the undirected Yao graph. \cgalFigureRef{f-y5} (b) gives an example of Yao graph with \f$k=5\f$.
Given a set \f$V\f$ of vertices on the plane, the directed Yao Graph with an integer parameter \f$k (k > 1)\f$ on \f$V\f$ is obtained as follows. For each vertex \f$u \in V\f$ starting from a given direction (e.g., the direction of positive \f$y\f$-axis), draw \f$k\f$ equally-spaced rays \f$l_0\f$, \f$l_1\f$, ..., \f$l_{k-1}\f$ originating from \f$u\f$ in clockwise order (see \cgalFigureRef{f-y5} (a)). These rays divide the plane into \f$k\f$ cones of angle \f$2\pi/k\f$, denoted by \f$ c(u, 1), c(u, 2), ..., c(u, k)\f$ respectively in clockwise order. To avoid overlapping at boundaries, it is assumed here that the area of \f$ c(u, i)\f$, where \f$ i=1, \ldots, k\f$, includes the ray \f$l_{i-1}\f$ but excludes the ray \f$l_{i\% k}\f$. In each cone of \f$u\f$, draw a directed edge from \f$u\f$ to its closest vertex by Euclidean distance in that cone. Ties are broken arbitrarily. These directed edges will form the edge set of the directed Yao graph on \f$V\f$. The undirected Yao Graph on \f$V\f$ is obtained by ignoring the directions of the edges. Note that if both edge \f$uv\f$ and \f$vu\f$ are in the directed Yao graph, only one edge \f$uv\f$ exists in the undirected Yao graph. \cgalFigureRef{f-y5} (b) gives an example of Yao graph with \f$k=5\f$.
\cgalFigureBegin{f-y5, Example-Y5.jpg}
Cones and an example of Yao Graph with \f$k=5\f$.