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:: is better than () as it then looks as it did with the old tools
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@ -50,30 +50,30 @@ class need not be specified and defaults to types and operations defined
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in the kernel in which the input point type is defined.
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Given a sequence of \f$ n\f$ input points with \f$ h\f$ extreme points,
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the function `convex_hull_2()` uses either the output-sensitive \f$ O(n h)\f$ algorithm of Bykat \cite b-chfsp-78
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the function `::convex_hull_2` uses either the output-sensitive \f$ O(n h)\f$ algorithm of Bykat \cite b-chfsp-78
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(a non-recursive version of the quickhull \cite bdh-qach-96 algorithm) or the algorithm of Akl and Toussaint, which requires \f$ O(n \log n)\f$ time
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in the worst case. The algorithm chosen depends on the kind of
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iterator used to specify the input points. These two algorithms are
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also available via the functions `ch_bykat()` and `ch_akl_toussaint()`,
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also available via the functions `::ch_bykat` and `::ch_akl_toussaint`,
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respectively. Also available are
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the \f$ O(n \log n)\f$ Graham-Andrew scan algorithm \cite a-aeach-79, \cite m-mdscg-84
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(`ch_graham_andrew()`),
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(`::ch_graham_andrew`),
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the \f$ O(n h)\f$ Jarvis march algorithm \cite j-ichfs-73
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(`ch_jarvis()`),
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(`::ch_jarvis`),
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and Eddy's \f$ O(n h)\f$ algorithm \cite e-nchap-77
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(`ch_eddy()`), which corresponds to the
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(`::ch_eddy`), which corresponds to the
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two-dimensional version of the quickhull algorithm.
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The linear-time algorithm of Melkman for producing the convex hull of
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simple polygonal chains (or polygons) is available through the function
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`ch_melkman()`.
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`::ch_melkman`.
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# Example using Graham-Andrew's Algorithm #
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In the following example a convex hull is constructed from point data read
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from standard input using `Graham_Andrew` algorithm. The resulting convex
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polygon is shown at the standard output console. The same results could be
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achieved by substituting the function `ch_graham_andrew()` by other
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function like `ch_bykat()`.
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achieved by substituting the function `::ch_graham_andrew` by other
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function like `::ch_bykat`.
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\cgalexample{Convex_hull_2/ch_from_cin_to_cout.cpp}
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@ -83,7 +83,7 @@ In addition to the functions for producing convex hulls, there are a
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number of functions for computing sets and sequences of points related
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to the convex hull.
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The functions `lower_hull_points_2()` and `upper_hull_points_2()`
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The functions `::lower_hull_points_2` and `::upper_hull_points_2`
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provide the computation of the counterclockwise
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sequence of extreme points on the lower hull and upper hull,
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respectively. The algorithm used in these functions is
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@ -92,13 +92,13 @@ which has worst-case running time of \f$ O(n \log n)\f$.
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There are also functions available for computing certain subsequences
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of the sequence of extreme points on the convex hull. The function
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`ch_jarvis_march()` generates the counterclockwise ordered subsequence of
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`::ch_jarvis_march` generates the counterclockwise ordered subsequence of
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extreme points between a given pair of points and
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`ch_graham_andrew_scan()` computes the sorted sequence of extreme points that are
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`::ch_graham_andrew_scan` computes the sorted sequence of extreme points that are
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not left of the line defined by the first and last input points.
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Finally, a set of functions (`ch_nswe_point()`, `ch_ns_point()`,
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`ch_we_point()`, `ch_n_point()`, `ch_s_point()`, `ch_w_point()`, `ch_e_point()`)
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Finally, a set of functions (`::ch_nswe_point`, `::ch_ns_point`,
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`::ch_we_point`, `::ch_n_point`, `::ch_s_point`, `::ch_w_point`, `::ch_e_point`)
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is provided for computing extreme points of a
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2D point set in the coordinate directions.
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@ -121,7 +121,7 @@ points projected into each of the three coordinate planes.
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# Convexity Checking #
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The functions `is_ccw_strongly_convex_2()` and `is_cw_strongly_convex_2()`
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The functions `::is_ccw_strongly_convex_2` and `::is_cw_strongly_convex_2`
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check whether a given sequence of 2D points forms a (counter)clockwise strongly
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convex polygon. These are used in postcondition
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testing of the two-dimensional convex hull functions.
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