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

CGAL 4.8 - 2D Polygon Partitioning

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PartitionIsValidTraits_2 Concept Reference
Concepts

## Definition

Requirements of a traits class that is used by `partition_is_valid_2`, `convex_partition_is_valid_2`, and `y_monotone_partition_is_valid_2` for testing if a given set of polygons are nonoverlapping and if their union is a polygon that is the same as a polygon represented by a given sequence of points. Note that the traits class for `partition_is_valid_2` may have to satisfy additional requirements if each partition polygon is to be tested for having a particular property; see, for example, the descriptions of the function `is_convex_2()` and the concept `YMonotonePartitionTraits_2` for the additional requirements for testing for convexity and \( y\)-monotonicity, respectively.

Has Models:
  `CGAL::Partition_is_valid_traits_2<Traits, PolygonIsValid>`

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 created by the partitioning function. More...
typedef unspecified_type | Is_valid
| A model of the concept `PolygonIsValid`
typedef unspecified_type | Less_xy_2
| Predicate object type that compares `Point_2`s lexicographically. More...
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 \( l\).

## Creation
Only a copy constructor is required.
| PartitionIsValidTraits_2 (PartitionIsValidTraits_2 tr)

## Operations
The following functions that create instances of the above predicate object types must exist.
Orientation_2 | is_valid_object ()
Less_xy_2 | less_xy_2_object ()
Left_turn_2 | left_turn_2_object ()
Orientation_2 | orientation_2_object ()

## Member Typedef Documentation

typedef unspecified_type PartitionIsValidTraits_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 PartitionIsValidTraits_2::Polygon_2

The polygon type created by the partitioning function.

This type should provide a nested type `Vertex_const_iterator` that is the type of the non-mutable iterator over the polygon vertices.