\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.4 - 2D Triangulation
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CGAL::Triangulation_euclidean_traits_xy_3< K > Class Template Reference
Traits Classes

#include <CGAL/Triangulation_euclidean_traits_xy_3.h>

Definition

The functionality of this class has been generalized to other packages than 2D triangulations.

The more general class Projection_traits_xy_3 can be found in the 2D and 3D Linear Geometric Kernel.

Deprecated:
The class Triangulation_euclidean_traits_xy_3 is a geometric traits class which allows to triangulate a terrain. This traits class is designed to build a two dimensional triangulation embedded in 3D space, i.e. a triangulated surface, such that its on the \( xy\) plane is a Delaunay triangulation. This is a usual construction for GIS terrains. Instead of really projecting the 3D points and maintaining a mapping between each point and its projection (which costs space and is error prone) the class Triangulation_euclidean_traits_xy_3 supplies geometric predicates that ignore the z-coordinate of the points.

The class is a model of the concept DelaunayTriangulationTraits_2 except that it does not provide the type and constructors required to build the dual Voronoi diagram. The class is also a model of the concept ConstrainedTriangulationTraits_2.

Parameters

The template parameter K has to be instantiated by a model of the Kernel concept. Triangulation_euclidean_traits_xy_3 uses types and predicates defined in K.

See Also
TriangulationTraits_2
DelaunayTriangulationTraits_2
CGAL::Triangulation_2<Traits,Tds>
CGAL::Delaunay_triangulation_2<Traits,Tds>

CGAL provides also predefined geometric traits class Triangulation_euclidean_traits_yz_3<K> and Triangulation_euclidean_traits_xz_3<K> to deal with projections on the xz- or the yz-plane, respectively.

See Also
CGAL/Triangulation_euclidean_traits_xz_3.h
CGAL/Triangulation_euclidean_traits_yz_3.h

Types

The following predicates and constructor types are provided

typedef Point_3< K > Point_2
 
typedef Segment_3< K > Segment_2
 
typedef Triangle_3< K > Triangle_2
 
typedef Line_3< K > Line_2
 
typedef unspecified_type Construct_segment_2
 A constructor object for Segment_2. More...
 
typedef unspecified_type Construct_triangle_2
 A constructor object for Triangle_2. More...
 
typedef unspecified_type Construct_line_2
 A constructor object for Line_2. More...
 
typedef unspecified_type Compute_squared_distance_2
 A construction object. More...
 
typedef unspecified_type Intersect_2
 A construction object. More...
 
typedef unspecified_type Compare_x_2
 Predicate object. More...
 
typedef unspecified_type Compare_y_2
 Predicate object. More...
 
typedef unspecified_type Orientation_2
 Predicate object. More...
 
typedef unspecified_type Side_of_oriented_circle_2
 Predicate object. More...
 
typedef unspecified_type Compare_distance_2
 Predicate object. More...
 

Creation

 Triangulation_euclidean_traits_xy_3 ()
 default constructor.
 
 Triangulation_euclidean_traits_xy_3 (Triangulation_euclidean_traits_xy_3 tr)
 Copy constructor.
 
Triangulation_euclidean_traits_xy_3 operator= (Triangulation_euclidean_traits_xy_3 tr)
 Assignment operator.
 

Access to predicate objects

The following access functions are provided

Construct_segment_2 construct_segment_2_object ()
 
Construct_triangle_2 construct_triangle_2_object ()
 
Construct_line_2 construct_line_2_object ()
 
Comparison_x_2 compare_x_2_object ()
 
Comparison_y_2 compare_y_2_object ()
 
Orientation_2 orientation_2_object ()
 
Side_of_oriented_circle_2 side_of_oriented_circle_2_object ()
 
Compare_distance_2 compare_distance_2_object ()
 
Intersect_2 intersect_2_object ()
 
Compute_squared_distance_2 compute_squared_distance_2_object ()
 

Member Typedef Documentation

template<typename K >
typedef unspecified_type CGAL::Triangulation_euclidean_traits_xy_3< K >::Compare_distance_2

Predicate object.

Provides the operator :

Comparison_result operator()(Point_2 p, Point_2 q, Point_2 r) which returns SMALLER, EQUAL or LARGER according to the distance between the projection of p and the projection of q being smaller, equal or larger than the distance between the projection of p and the projection of r.

template<typename K >
typedef unspecified_type CGAL::Triangulation_euclidean_traits_xy_3< K >::Compare_x_2

Predicate object.

Provides the operator :

Comparison_result operator()(Point_2 p, Point_2 q)

which returns SMALLER, EQUAL or LARGER according to the \( x\)-ordering of points p and q.

template<typename K >
typedef unspecified_type CGAL::Triangulation_euclidean_traits_xy_3< K >::Compare_y_2

Predicate object.

Provides the operator :

Comparison_result operator()(Point_2 p, Point_2 q)

which returns (SMALLER, EQUAL or LARGER) according to the \( y\)-ordering of points p and q.

template<typename K >
typedef unspecified_type CGAL::Triangulation_euclidean_traits_xy_3< K >::Compute_squared_distance_2

A construction object.

Provides the operator :

RT operator()(Line_2 l, Point_2 p); which returns the squared distance between the projection of p and the projection of l.

template<typename K >
typedef unspecified_type CGAL::Triangulation_euclidean_traits_xy_3< K >::Construct_line_2

A constructor object for Line_2.

Provides :

Line_2 operator()(Point_2 p,Point_2 q),

which constructs a line from two points.

template<typename K >
typedef unspecified_type CGAL::Triangulation_euclidean_traits_xy_3< K >::Construct_segment_2

A constructor object for Segment_2.

Provides :

Segment_2 operator()(Point_2 p,Point_2 q),

which constructs a segment from two points.

template<typename K >
typedef unspecified_type CGAL::Triangulation_euclidean_traits_xy_3< K >::Construct_triangle_2

A constructor object for Triangle_2.

Provides :

Triangle_2 operator()(Point_2 p,Point_2 q,Point_2 r ),

which constructs a triangle from three points.

template<typename K >
typedef unspecified_type CGAL::Triangulation_euclidean_traits_xy_3< K >::Intersect_2

A construction object.

Provides the operator :

Object_2 operator()(Segment_2 s1, Segment_2 s2); which returns the intersection of the projection of s1 and the projection of s2 embedded in 3D. If the intersection is a segment, the z-coordinates of its extremities is 0. If the intersection is a point p, let p1 and p2 be the points on s1 and s2 respectively, such that their projections are p. The point returned is the middle of the segment p1p2.

Precondition
The projection of s1 and the projection of s2 are non-degenerate 2D segments.
template<typename K >
typedef unspecified_type CGAL::Triangulation_euclidean_traits_xy_3< K >::Orientation_2

Predicate object.

Provides the operator :

Orientation operator()(Point_2 p, Point_2 q, Point_2 r)

which returns LEFT_TURN, RIGHT_TURN or COLLINEAR according to the position of the projection of \( r\) with respect to the projection of the oriented line pq.

template<typename K >
typedef unspecified_type CGAL::Triangulation_euclidean_traits_xy_3< K >::Side_of_oriented_circle_2

Predicate object.

Provides the operator : Oriented_side operator()(Point_2 p, Point_2 q, Point_2 r, Point_2 s) which takes four points \( p, q, r, s\) as arguments and returns ON_POSITIVE_SIDE, ON_NEGATIVE_SIDE or, ON_ORIENTED_BOUNDARY according to the position of the projection of points with respect to the oriented circle through the projections of \( p,q\) and \( r\).