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% | Reference manual page: PartitionTraits_2.tex
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% | 10.05.2000   Susan Hert
% | Package: Decomposition_2
% | 
% |
% +------------------------------------------------------------------------+


\begin{ccRefConcept}{PartitionTraits_2}

\ccDefinition
  
The polygon partitioning functions are each parameterized by a traits class 
that defines the primitives used in the algorithms.  Many requirements are 
common
to all traits classes.  The concept \ccRefName\ defines this common set of
requirements.  

\ccTypes

\ccNestedType{Point_2}{The point type on which the partitioning algorithm operates.}

\ccNestedType{Polygon_2}{The polygon type to be created by the partitioning 
algorithm. For testing the validity postcondition of the partition, this 
type should provide a nested type \ccc{Vertex_const_iterator} that is the
type of the iterator over the polygon vertices and member functions
\ccc{Vertex_const_iterator vertices_begin()} and
\ccc{Vertex_const_iterator vertices_end()}.}%
\ccIndexSubitem[C]{approx_convex_partition_2}{postconditions}
\ccIndexSubitem[C]{greene_approx_convex_partition_2}{postconditions}
\ccIndexSubitem[C]{optimal_convex_partition_2}{postconditions}
\ccIndexSubitem[C]{y_monotone_partition_2}{postconditions}

\ccNestedType{Less_xy_2}{
Predicate object type that compares \ccc{Point_2}s lexicographically.
Must provide \ccc{bool operator()(Point_2 p, Point_2 q)} where \ccc{true}
is returned iff $p <_{xy} q$.
We have $p<_{xy}q$, iff $p_x < q_x$ or $p_x = q_x$ and $p_y < q_y$,
where $p_x$ and $p_y$ denote the $x$ and $y$ coordinates of point $p$, 
respectively.
}

\ccNestedType{Less_yx_2}{
Same as \ccc{Less_xy_2} with the roles of $x$ and $y$ interchanged.}


\ccNestedType{Left_turn_2}{
Predicate object type that provides 
\ccc{bool operator()(Point_2 p,Point_2 q,Point_2 r)}, which
returns \ccc{true} iff \ccc{r} lies to the left of the 
oriented line through \ccc{p} and \ccc{q}.}

\ccNestedType{Orientation_2}{Predicate object type that provides
\ccc{CGAL::Orientation operator()(Point_2 p, Point_2 q, Point_2 r)} that
returns \ccStyle{CGAL::LEFT_TURN}, if $r$ lies to the left of the oriented 
line $l$ defined by $p$ and $q$, returns \ccStyle{CGAL::RIGHT_TURN} if $r$ 
lies to the right of $l$, and returns \ccStyle{CGAL::COLLINEAR} if $r$ lies
on $l$.}

\ccNestedType{Compare_y_2}{Predicate object type that provides
\ccc{CGAL::Comparision_result operator()(Point_2 p, Point_2 q)} to compare
the $y$ values of two points.  The operator must return
\ccc{CGAL::SMALLER} if $p_y < q_y$, \ccc{CGAL::LARGER} if $p_y > q_y$ and
\ccc{CGAL::EQUAL} if $p_y = q_y$.}

\ccNestedType{Compare_x_2}{The same as \ccc{Compare_y_2}, except that $x$ 
coordinates are compared instead of $y$.}

%\ccNestedType{Construct_line_2}{Predicate object type that provides
%\ccc{Line_2 operator()(Point_2 p, Point_2 q)}, which constructs and
%returns the line defined by the points $p$ and $q$.}


\ccCreation
\ccCreationVariable{traits}  %% choose variable name

A copy constructor and default constructor are required.

\ccConstructor{PartitionTraits_2()}{}

\ccConstructor{PartitionTraits_2(PartitionTraits_2& tr)}{}

\ccOperations

The following functions that create instances of the above predicate object
types must exist.

\ccMethod{Less_yx_2 less_yx_2_object();}{}

\ccMethod{Less_xy_2 less_xy_2_object();}{}

\ccMethod{Left_turn_2 left_turn_2_object();}{}

\ccMethod{Orientation_2 orientation_2_object();}{}

\ccMethod{Compare_y_2 compare_y_2_object();}{}

\ccMethod{Compare_x_2 compare_x_2_object();}{}

%\ccMethod{Construct_line_2 construct_line_2_object();}{}

%The following functions are required by the functions \ccc{is_simple_2} and
%\ccc{orientation_2}, which are used to test the simplicity and CCW order 
%preconditions of the partitioning functions. 
%
%\ccMemberFunction{
%    Comparison_result compare_x(const Point_2 &p, const Point_2 &q) const;
%}
%{
%\lcTex{
%Returns
%$
%  \left\{
%  \begin{array}{lll}
%     \ccStyle{SMALLER} & \mbox{if} & p_x < q_x \\
%     \ccStyle{EQUAL}   & \mbox{if} & p_x = q_x \\
%     \ccStyle{LARGER}  & \mbox{if} & p_x > q_x
%  \end{array}
%  \right.%}
%$
%}
%\lcHtml{
%Returns SMALLER if p_x < q_x, EQUAL if p_x = q_x and LARGER if p_x > q_x
%}
%}
%
%
%\ccMemberFunction{
%    Comparison_result compare_y(const Point_2 &p, const Point_2 &q) const;
%}
%{
%\lcTex{
%Returns
%$
%  \left\{
%  \begin{array}{lll}
%     \ccStyle{SMALLER} & \mbox{if} & p_y < q_y \\
%     \ccStyle{EQUAL}   & \mbox{if} & p_y = q_y \\
%     \ccStyle{LARGER}  & \mbox{if} & p_y > q_y
%  \end{array}
%  \right.%}
%$
%}
%\lcHtml{
%Returns SMALLER if p_y < q_y, EQUAL if p_y = q_y and LARGER if p_y > q_y
%}
%}
%
%\ccMemberFunction{
%    FT cross_product_2(const Vector_2& p, const Vector_2& q) const;
%}
%{
%Returns $p_x q_y - p_y q_x$.
%}
%
%\ccMemberFunction{
%    bool do_intersect(const Point_2& p1,
%                      const Point_2& q1,
%                      const Point_2& p2,
%                      const Point_2& q2) const;
%}
%{
%Returns \ccc{true} iff the segments \ccStyle{[p1,q1]} and \ccStyle{[p2,q2]}
%intersect.
%}
%
%\ccMemberFunction{
%    bool have_equal_direction(const Vector_2& v1,
%                              const Vector_2& v2 ) const;
%}
%{
%Returns \ccc{true} iff the vectors \ccStyle{v1} and \ccStyle{v2} have the
%same direction.
%}
%
%\ccMemberFunction{
%    bool is_negative(const FT& x) const;
%}
%{
%Returns \ccStyle{true} iff \ccStyle{x<0}.
%}
%
%\ccMemberFunction{
%    bool lexicographically_yx_smaller_or_equal(const Point_2& p,
%                                               const Point_2& q) const;
%}
%{
%Returns \ccc{true} iff $p_y \leq q_y$ or $p_y = q_y$ and $p_x \leq q_x$.
%}
%
%\ccMethod{
%    Orientation
%    orientation(const Point_2& p, const Point_2& q, const Point_2& r) const;
%}
%{
%Returns \ccStyle{LEFT_TURN}, if $r$ lies to the left of the oriented
%line $l$ defined by $p$ and $q$, returns \ccStyle{RIGHT_TURN} if $r$
%lies to the right of $l$, and returns \ccStyle{COLLINEAR} if $r$ lies
%on $l$.
%%}



\ccHasModels

\ccRefIdfierPage{CGAL::Partition_traits_2<R>}

\ccSeeAlso

\ccRefIdfierPage{CGAL::approx_convex_partition_2} \\
\ccRefIdfierPage{CGAL::greene_approx_convex_partition_2} \\
\ccRefIdfierPage{CGAL::optimal_convex_partition_2} \\
\ccRefIdfierPage{CGAL::y_monotone_partition_2}

\end{ccRefConcept}

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