\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)

CGAL 4.8 - 2D Polygon Partitioning

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PartitionTraits_2 Concept Reference
Concepts

## Definition

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 `PartitionTraits_2` defines this common set of requirements.

Has Models:
  `CGAL::Partition_traits_2<R>`

See Also
  `CGAL::approx_convex_partition_2()`
  `CGAL::greene_approx_convex_partition_2()`
  `CGAL::optimal_convex_partition_2()`
  `CGAL::y_monotone_partition_2()`

## Types
typedef unspecified_type | Point_2
| The point type on which the partitioning algorithm operates.
typedef unspecified_type | Polygon_2
| The polygon type to be created by the partitioning algorithm. More...
typedef unspecified_type | Less_xy_2
| Predicate object type that compares `Point_2`s lexicographically. More...
typedef unspecified_type | Less_yx_2
| Same as `Less_xy_2` with the roles of \( x\) and \( y\) interchanged.
typedef unspecified_type | Left_turn_2
| Predicate object type that provides `bool operator()(Point_2 p,Point_2 q,Point_2 r)`, which returns `true` iff `r` lies to the left of the oriented line through `p` and `q`.
typedef unspecified_type | Orientation_2
| Predicate object type that provides `CGAL::Orientation operator()(Point_2 p, Point_2 q, Point_2 r)` that returns `CGAL::LEFT_TURN`, if \( r\) lies to the left of the oriented line \( l\) defined by \( p\) and \( q\), returns `CGAL::RIGHT_TURN` if \( r\) lies to the right of \( l\), and returns `CGAL::COLLINEAR` if \( r\) lies on \( l\).
typedef unspecified_type | Compare_y_2
| Predicate object type that provides `CGAL::Comparision_result operator()(Point_2 p, Point_2 q)` to compare the \( y\) values of two points. More...
typedef unspecified_type | Compare_x_2
| The same as `Compare_y_2`, except that \( x\) coordinates are compared instead of \( y\).

## Creation
A copy constructor and default constructor are required.
| PartitionTraits_2 ()
| PartitionTraits_2 (PartitionTraits_2 tr)

## Operations
The following functions that create instances of the above predicate object types must exist.
Less_yx_2 | less_yx_2_object () const
Less_xy_2 | less_xy_2_object () const
Left_turn_2 | left_turn_2_object () const
Orientation_2 | orientation_2_object () const
Compare_y_2 | compare_y_2_object () const
Compare_x_2 | compare_x_2_object () const

## Member Typedef Documentation

typedef unspecified_type PartitionTraits_2::Compare_y_2

Predicate object type that provides `CGAL::Comparision_result operator()(Point_2 p, Point_2 q)` to compare the \( y\) values of two points.

The operator must return `CGAL::SMALLER` if \( p_y < q_y\), `CGAL::LARGER` if \( p_y > q_y\) and `CGAL::EQUAL` if \( p_y = q_y\).

typedef unspecified_type PartitionTraits_2::Less_xy_2

Predicate object type that compares `Point_2`s lexicographically.

Must provide `bool operator()(Point_2 p, Point_2 q)` where `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.

typedef unspecified_type PartitionTraits_2::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 `Vertex_const_iterator` that is the type of the iterator over the polygon vertices and member functions `Vertex_const_iterator vertices_begin()` and `Vertex_const_iterator vertices_end()`.