CGAL::Arr_algebraic_segment_traits_2< Coefficient > Class Template Reference

#include <CGAL/Arr_algebraic_segment_traits_2.h>

Definition

The traits class Arr_algebraic_segment_traits_2 is a model of the ArrangementTraits_2 concept that handles planar algebraic curves of arbitrary degree, and \( x\)-monotone of such curves.

A planar (real) algebraic curve is the vanishing set of a polynomial in two variables, that is, the curve is defined by the defining equation

\[ f(x):=\sum_{i+j\leq n} a_{ij} x^i y^j =0, \]

where \( n\) is the degree of the curve.

The traits class allows the construction of algebraic curves, by specifying their implicit equation. \( x\)-monotone and vertical segments of a curve can also be defined; unbounded curves and segments are supported. The template parameter Coefficient defines the innermost coefficient type of the polynomials. Currently, the types leda::integer and CORE::BigInt are supported as well as any instance of CGAL::Sqrt_extension that is instantiated with one of the integral types above.

Is Model Of:

ArrangementTraits_2

Examples:

Arrangement_on_surface_2/algebraic_curves.cpp, and Arrangement_on_surface_2/algebraic_segments.cpp.

Classes
class	Construct_curve_2
class	Construct_point_2
class	Construct_x_monotone_segment_2
class	Curve_2
	Models the ArrangementTraits_2::Curve_2 concept. More...
class	Point_2
	Models the ArrangementBasicTraits_2::Point_2 concept. More...
class	X_monotone_curve_2
	Models the ArrangementBasicTraits_2::X_monotone_curve_2 concept. More...

Types
enum	Site_of_point { POINT_IN_INTERIOR = 0, MIN_ENDPOINT = -1, MAX_ENDPOINT = 1 }
	Value to specify whether a point should be in the interior of a segment, or its minimal point, or its maximal point in lexicographic order. More...
typedef unspecified_type	Polynomial_2
	the type for bivariate polynomials, with innermost coefficient type Coefficient. More...
typedef unspecified_type	Algebraic_kernel_1
	model for the concept AlgebraicKernel_1
typedef unspecified_type	Algebraic_real_1
	represents coordinates of points. More...
typedef unspecified_type	Bound
	Typedef from Algebraic_kernel_1::Bound

Accessing functor objects
Construct_curve_2	construct_curve_2_object () const
Construct_point_2	construct_point_2_object () const
Construct_x_monotone_segment_2	construct_x_monotone_segment_2_object () const

Member Typedef Documentation

Algebraic_real_1

template<typename Coefficient >

typedef unspecified_type CGAL::Arr_algebraic_segment_traits_2< Coefficient >::Algebraic_real_1

represents coordinates of points.

Typedef from Algebraic_kernel_1::Algebraic_real_1

Polynomial_2

template<typename Coefficient >

typedef unspecified_type CGAL::Arr_algebraic_segment_traits_2< Coefficient >::Polynomial_2

the type for bivariate polynomials, with innermost coefficient type Coefficient.

Constitutes a model of the concept Polynomial_d with two variables.

See also

CGAL::Polynomial_d

Member Enumeration Documentation

Site_of_point

template<typename Coefficient >

enum CGAL::Arr_algebraic_segment_traits_2::Site_of_point

Value to specify whether a point should be in the interior of a segment, or its minimal point, or its maximal point in lexicographic order.

Enumerator
POINT_IN_INTERIOR
MIN_ENDPOINT
MAX_ENDPOINT