namespace CGAL {
/*!

\mainpage User Manual 
\anchor Chapter_2D_Convex_Hulls_and_Extreme_Points
\anchor chapconvexhull2
\cgalAutoToc
\authors Susan Hert and Stefan Schirra

\section Convex_hull_2Introduction Introduction

A subset \f$ S \subseteq \mathbb{R}^2\f$ is convex if for any two points \f$ p\f$ and \f$ q\f$
in the set the line segment with endpoints \f$ p\f$ and \f$ q\f$ is contained
in \f$ S\f$. The convex hull of a set \f$ S\f$ 
is the smallest convex set containing
\f$ S\f$. The convex hull of a set of points \f$ P\f$ is a convex 
polygon with vertices in \f$ P\f$. A point in \f$ P\f$ is an extreme point 
(with respect to \f$ P\f$) if it is a vertex of 
the convex hull of \f$ P\f$. A set of points is said to be strongly convex 
if it consists of only extreme points.

This chapter describes the functions provided in
\cgal for producing convex hulls in two dimensions as well as
functions for checking if sets of points are strongly convex are not. 
There are also a number of functions described for computing particular 
extreme points and subsequences of hull points, such as the lower and 
upper hull of a set of points.

\image html saarhull.png
\image latex saarhull.png

\section secconvex_hull_2 Convex Hull 

\cgal provides implementations of several classical algorithms for
computing the counterclockwise sequence of extreme points for a set of 
points in two dimensions (<I>i.e.</I>, the counterclockwise sequence 
of points on the convex hull). The algorithms have different asymptotic
running times and require slightly different sets of geometric primitives. 
Thus you may choose the algorithm that best fits your setting.

Each of the convex hull functions presents the same interface to the
user. That is, the user provides a pair of iterators, `first`
and `beyond`, an output iterator `result`, and a traits class
`traits`. The points in the range [`first`, `beyond`) define
the input points whose convex hull is to be computed. The counterclockwise
sequence of extreme points is written to the sequence starting at position
`result`, and the past-the-end iterator for the resulting set of
points is returned. The traits classes for the functions specify the types
of the input points and the geometric primitives that are required by
the algorithms. All functions provide an interface in which this
class need not be specified and defaults to types and operations 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,
the function `convex_hull_2()` uses either the output-sensitive \f$ O(n h)\f$ algorithm of Bykat \cgalCite{b-chfsp-78}
(a non-recursive version of the quickhull \cgalCite{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 
iterator used to specify the input points. These two algorithms are
also available via the functions `ch_bykat()` and `ch_akl_toussaint()`,
respectively. Also available are 
the \f$ O(n \log n)\f$ Graham-Andrew scan algorithm \cgalCite{a-aeach-79}, \cgalCite{m-mdscg-84} 
(`ch_graham_andrew()`), 
the \f$ O(n h)\f$ Jarvis march algorithm \cgalCite{j-ichfs-73}
(`ch_jarvis()`),
and Eddy's \f$ O(n h)\f$ algorithm \cgalCite{e-nchap-77}
(`ch_eddy()`), which corresponds to the 
two-dimensional version of the quickhull algorithm.
The linear-time algorithm of Melkman for producing the convex hull of 
simple polygonal chains (or polygons) is available through the function
`ch_melkman()`. 

\section Convex_hull_2Example Example using Graham-Andrew's Algorithm

In the following example a convex hull is constructed from point data read 
from standard input using `Graham_Andrew` algorithm. The resulting convex 
polygon is shown at the standard output console. The same results could be 
achieved by substituting the function `ch_graham_andrew()` by other 
functions such as `ch_bykat()`.

\cgalExample{Convex_hull_2/ch_from_cin_to_cout.cpp}

\section Convex_hull_2Extreme Extreme Points and Hull Subsequences

In addition to the functions for producing convex hulls, there are a
number of functions for computing sets and sequences of points related
to the convex hull. 

The functions `lower_hull_points_2()` and `upper_hull_points_2()`
provide the computation of the counterclockwise 
sequence of extreme points on the lower hull and upper hull,
respectively. The algorithm used in these functions is
Andrew's variant of Graham's scan algorithm \cgalCite{a-aeach-79}, \cgalCite{m-mdscg-84},
which has worst-case running time of \f$ O(n \log n)\f$.

There are also functions available for computing certain subsequences 
of the sequence of extreme points on the convex hull. The function
`ch_jarvis_march()` generates the counterclockwise ordered subsequence of
extreme points between a given pair of points and
`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.

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()`)
is provided for computing extreme points of a 
2D point set in the coordinate directions. 

\section Convex_hull_2Traits Traits Classes

Each of the functions used to compute convex hulls or extreme points
is parameterized by a traits class, which specifies the types and geometric
primitives to be used in the computation. There are several implementations 
of 2D traits classes provided in the library. The class
`Convex_hull_traits_2` corresponds to the default traits class that provides the types and 
predicates presented in the 2-dimensional \cgal kernel in which the input 
points lie. The class `Convex_hull_constructive_traits_2` is a second traits class
 based on \cgal primitives but differs from `Convex_hull_traits_2` in that some of its primitives reuse
intermediate results to speed up computation.

In addition, the 2D and 3D Linear Geometric Kernel provides three projective traits classes 
(`Projection_traits_xy_3`, `Projection_traits_xz_3`, and `Projection_traits_yz_3`), 
which may be used to compute the convex hull of a set of three-dimensional
points projected into each of the three coordinate planes.

\section Convex_hull_2Convexity Convexity Checking

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
convex polygon. These are used in postcondition
testing of the two-dimensional convex hull functions.

*/ 
} /* namespace CGAL */

