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:: is better than () as it then looks as it did with the old tools

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Andreas Fabri 2012-08-20 10:39:50 +00:00
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Convex_hull_2/doc/Convex_hull_2/Convex_hull_2.txt
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@ -50,30 +50,30 @@ class need not be specified and defaults to types and operations defined
in the kernel in which the input point type is defined. in the kernel in which the input point type is defined.
Given a sequence of \f$ n\f$ input points with \f$ h\f$ extreme points, Given a sequence of \f$ n\f$ input points with \f$ h\f$ extreme points,
the function `convex_hull_2()` uses either the output-sensitive \f$ O(n h)\f$ algorithm of Bykat \cite b-chfsp-78 the function `::convex_hull_2` uses either the output-sensitive \f$ O(n h)\f$ algorithm of Bykat \cite b-chfsp-78
(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 (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
in the worst case. The algorithm chosen depends on the kind of in the worst case. The algorithm chosen depends on the kind of
iterator used to specify the input points. These two algorithms are iterator used to specify the input points. These two algorithms are
also available via the functions `ch_bykat()` and `ch_akl_toussaint()`, also available via the functions `::ch_bykat` and `::ch_akl_toussaint`,
respectively. Also available are respectively. Also available are
the \f$ O(n \log n)\f$ Graham-Andrew scan algorithm \cite a-aeach-79, \cite m-mdscg-84 the \f$ O(n \log n)\f$ Graham-Andrew scan algorithm \cite a-aeach-79, \cite m-mdscg-84
(`ch_graham_andrew()`), (`::ch_graham_andrew`),
the \f$ O(n h)\f$ Jarvis march algorithm \cite j-ichfs-73 the \f$ O(n h)\f$ Jarvis march algorithm \cite j-ichfs-73
(`ch_jarvis()`), (`::ch_jarvis`),
and Eddy's \f$ O(n h)\f$ algorithm \cite e-nchap-77 and Eddy's \f$ O(n h)\f$ algorithm \cite e-nchap-77
(`ch_eddy()`), which corresponds to the (`::ch_eddy`), which corresponds to the
two-dimensional version of the quickhull algorithm. two-dimensional version of the quickhull algorithm.
The linear-time algorithm of Melkman for producing the convex hull of The linear-time algorithm of Melkman for producing the convex hull of
simple polygonal chains (or polygons) is available through the function simple polygonal chains (or polygons) is available through the function
`ch_melkman()`. `::ch_melkman`.
# Example using Graham-Andrew's Algorithm # # Example using Graham-Andrew's Algorithm #
In the following example a convex hull is constructed from point data read In the following example a convex hull is constructed from point data read
from standard input using `Graham_Andrew` algorithm. The resulting convex from standard input using `Graham_Andrew` algorithm. The resulting convex
polygon is shown at the standard output console. The same results could be polygon is shown at the standard output console. The same results could be
achieved by substituting the function `ch_graham_andrew()` by other achieved by substituting the function `::ch_graham_andrew` by other
function like `ch_bykat()`. function like `::ch_bykat`.
\cgalexample{Convex_hull_2/ch_from_cin_to_cout.cpp} \cgalexample{Convex_hull_2/ch_from_cin_to_cout.cpp}
@ -83,7 +83,7 @@ In addition to the functions for producing convex hulls, there are a
number of functions for computing sets and sequences of points related number of functions for computing sets and sequences of points related
to the convex hull. to the convex hull.
The functions `lower_hull_points_2()` and `upper_hull_points_2()` The functions `::lower_hull_points_2` and `::upper_hull_points_2`
provide the computation of the counterclockwise provide the computation of the counterclockwise
sequence of extreme points on the lower hull and upper hull, sequence of extreme points on the lower hull and upper hull,
respectively. The algorithm used in these functions is respectively. The algorithm used in these functions is
@ -92,13 +92,13 @@ which has worst-case running time of \f$ O(n \log n)\f$.
There are also functions available for computing certain subsequences There are also functions available for computing certain subsequences
of the sequence of extreme points on the convex hull. The function of the sequence of extreme points on the convex hull. The function
`ch_jarvis_march()` generates the counterclockwise ordered subsequence of `::ch_jarvis_march` generates the counterclockwise ordered subsequence of
extreme points between a given pair of points and extreme points between a given pair of points and
`ch_graham_andrew_scan()` computes the sorted sequence of extreme points that are `::ch_graham_andrew_scan` computes the sorted sequence of extreme points that are
not left of the line defined by the first and last input points. not left of the line defined by the first and last input points.
Finally, a set of functions (`ch_nswe_point()`, `ch_ns_point()`, Finally, a set of functions (`::ch_nswe_point`, `::ch_ns_point`,
`ch_we_point()`, `ch_n_point()`, `ch_s_point()`, `ch_w_point()`, `ch_e_point()`) `::ch_we_point`, `::ch_n_point`, `::ch_s_point`, `::ch_w_point`, `::ch_e_point`)
is provided for computing extreme points of a is provided for computing extreme points of a
2D point set in the coordinate directions. 2D point set in the coordinate directions.
@ -121,7 +121,7 @@ points projected into each of the three coordinate planes.
# Convexity Checking # # Convexity Checking #
The functions `is_ccw_strongly_convex_2()` and `is_cw_strongly_convex_2()` The functions `::is_ccw_strongly_convex_2` and `::is_cw_strongly_convex_2`
check whether a given sequence of 2D points forms a (counter)clockwise strongly check whether a given sequence of 2D points forms a (counter)clockwise strongly
convex polygon. These are used in postcondition convex polygon. These are used in postcondition
testing of the two-dimensional convex hull functions. testing of the two-dimensional convex hull functions.