CGAL::Arr_circle_segment_traits_2<Kernel>

Definition

The class Arr_circle_segment_traits_2<Kernel> is a model of the ArrangementTraits_2 concept and can be used to construct and maintain arrangements of circular arcs and line segments.

The traits class must be instantiated with a geometric kernel, such that the supporting circles of the circular arcs are of type Kernel::Circle_2 and the supporting lines of the line segments are of type Kernel::Line_2. Thus, the coordinates of the center of supporting circles, and its squared radius are of type Kernel::FT, which should be an exact rational number-type; similarly, the coefficients of each supporting line $$ax + by + c = 0 are also of type Kernel::FT. Note however that the intersection point between two such arcs do not have rational coordinates in general. For this reason, we do not require the endpoints of the input arcs and segments to have rational coordinates.

The nested Point_2 type defined by the traits class is therefore different than the Kernel::Point_2 type. Its coordinates are of type CoordNT, and represent real numbers obtained from solving quadratic equations with rational coordinates. A number of type CoordNT can therefore be expressed as $$+ sqrt(), where $$, $$ and $$ are all rational numbers. The definition of the curve and $$x-monotone curve types nested in the traits class are detailed below.

#include <CGAL/Arr_circle_segment_traits_2.h>

Is Model for the Concepts

Class Arr_circle_segment_traits_2$$<Kernel$$>::CoordNT

The CoordNT number-type nested within the traits class represents an algebraic number of degree 2; it can be represented as $$+ sqrt(), where $$, $$ and $$ are all rational numbers of type Kernel::FT.

Types

Creation

Arr_circle_segment_traits_2<Kernel>::CoordNT x;
	creates a variable whose value is $$0.
Arr_circle_segment_traits_2<Kernel>::CoordNT x ( Rational alpha);
	creates a variable whose value is the rational number $$.
Arr_circle_segment_traits_2<Kernel>::CoordNT x ( Rational alpha, Rational beta, Rational gamma);
	creates a variable whose value is the algebraic number $$+ sqrt().

Access Functions

bool	x.is_rational ()	determines whether x is rational.
Rational	x.alpha ()	returns $$.
Rational	x.beta ()	returns $$ (0 if x is rational).
Rational	x.gamma ()	returns $$ (0 if x is rational).

Operators

The CoordNT number-type supports all arithmetic operations with rational numbers of type Kernel::FT. For example, it is possible to compute x + q, q - x, x += q, etc. where q is rational.

The global functions CGAL::sign(x), CGAL::square(x), CGAL::to_double(x) and CGAL::compare(x,y), where x and y are of type CoordNT, are also supported.

It is also possible to call CGAL::to_double(x) to convert x to a rational number, with the precondition that it is indeed rational.

Class Arr_circle_segment_traits_2$$<Kernel$$>::Point_2

The Point_2 number-type nested within the traits class represents a Cartesian point whose coordinates are algebraic numbers of type CoordNT.

Types

Creation

Access Functions

Class Arr_circle_segment_traits_2$$<Kernel$$>::Curve_2

The Curve_2 class nested within the traits class can represent arbitrary circular arcs, full circles and line segments and support their construction in various ways. The copy and default constructor as well as the assignment operator are provided. In addition, an operator<< for the curves is defined for standard output streams.

Creation

Access Functions

Class Arr_circle_segment_traits_2$$<Kernel$$>::X_monotone_curve_2

The X_monotone_curve_2 class nested within the traits class can represent $$x-monotone and line segments (which are always weakly $$x-monotone). The copy and default constructor as well as the assignment operator are provided. In addition, an operator<< for the curves is defined for standard output streams.

Creation

Access Functions