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

CGAL 4.8 - 2D Arrangements

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CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2 Class Reference

`#include <CGAL/Arr_conic_traits_2.h>`

## Definition

The `Curve_2` class nested within the conic-arc traits can represent arbitrary conic arcs and support their construction in various ways.

The copy and default constructor as well as the assignment operator are provided for conic arcs. In addition, an `operator<<` for the curves is defined for standard output streams.

## Creation
| Curve_2 (const typename RatKernel::Segment_2 &seg)
| constructs an arc corresponding to the line segment `seg`.
| Curve_2 (const typename RatKernel::Circle_2 &circ)
| constructs an arc corresponding to the full circle `circ` (note that this circle has a center point with rational coordinates and rational squared radius).
| Curve_2 (const typename RatKernel::Circle_2 &circ, Orientation o, const Point_2 &ps, const Point_2 &pt)
| constructs a circular arc supported by the circle `circ`, going in the given orientation `o` from the source point `ps` to its target point `pt`. More...
| Curve_2 (const typename RatKernel::Point_2 &p1, const typename RatKernel::Point_2 &p2, const typename RatKernel::Point_2 &p3)
| constructs a circular arc going from `p1` (its source point) through `p2` to `p3` (its target point). More...
| Curve_2 (const Rational &r, const Rational &s, const Rational &t, const Rational &u, const Rational &v, const Rational &w)
| constructs a conic arc that corresponds to the full conic curve \( r x^2 + s y^2 + t x y + u x + v y + w = 0\). More...
| Curve_2 (const Rational &r, const Rational &s, const Rational &t, const Rational &u, const Rational &v, const Rational &w, Orientation o, const Point_2 &ps, const Point_2 &pt)
| constructs a conic arc supported by the conic curve \( r x^2 + s y^2 + t x y + u x + v y + w = 0\), going in the given orientation `o` from the source point `ps` to its target point `pt`. More...
| Curve_2 (const typename RatKernel::Point_2 &p1, const typename RatKernel::Point_2 &p2, const typename RatKernel::Point_2 &p3, const typename RatKernel::Point_2 &p4, const typename RatKernel::Point_2 &p5)
| constructs a conic arc going from `p1` (its source point) through `p2`, `p3` and `p4` (in this order) to `p5` (its target point). More...
| Curve_2 (const Rational &r, const Rational &s, const Rational &t, const Rational &u, const Rational &v, const Rational &w, Orientation o, const Point_2 &app_ps, const Rational &r1, const Rational &s1, const Rational &t1, const Rational &u1, const Rational &v1, const Rational &w1, const Point_2 &app_pt, const Rational &r2, const Rational &s2, const Rational &t2, const Rational &u2, const Rational &v2, const Rational &w2)
| constructs a conic arc supported by the conic curve \( r x^2 + s y^2 + t x y + u x + v y + w = 0\), going in the given orientation `o` from its source point to its target Point. More...

## Access Functions
bool | is_valid () const
| indicates whether `a` is a valid conic arc. More...
bool | is_x_monotone () const
| determines whether the arc is \( x\)-monotone, namely each vertical line intersects it at most once. More...
bool | is_y_monotone () const
| determines whether the arc is \( y\)-monotone, namely each horizontal line intersects it at most once. More...
bool | is_full_conic () const
| indicates whether the arc represents a full conic curve (en ellipse or a circle).
The six following methods return the coefficients of the supported conic, after their conversion to integer number (represented by the `Integer` type of the `NtTraits` class):
const NtTraits::Integer & | r () const
| returns the coefficient of \( x^2\).
const NtTraits::Integer & | s () const
| returns the coefficient of \( t^2\).
const NtTraits::Integer & | t () const
| returns the coefficient of \( x y\).
const NtTraits::Integer & | u () const
| returns the coefficient of \( x\).
const NtTraits::Integer & | v () const
| returns the coefficient of \( y\).
const NtTraits::Integer & | w () const
| returns the free coefficient.
const Point_2 & | source () const
| returns the source point of the arc. More...
const Point_2 & | target () const
| returns the target point of the arc. More...
Orientation | orientation () const
| returns the orientation of the arc.
Bbox_2 | bbox () const
| return a bounding box of the arc `a`.

## Operations
void | set_source (const Point_2 &ps)
| sets a new source point for the conic arc. More...
void | set_target (const Point_2 &pt)
| sets a new target point for the conic arc. More...

## Constructor & Destructor Documentation

template<typename RatKernel , typename AlgKernel , typename NtTraits >

CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::Curve_2 | ( | const typename RatKernel::Circle_2 & | circ,
|  | Orientation | o,
|  | const Point_2 & | ps,
|  | const Point_2 & | pt
| ) |  |

constructs a circular arc supported by the circle `circ`, going in the given orientation `o` from the source point `ps` to its target point `pt`.

Precondition
  `ps` and `pt` both lie on the circle `circ`.
  `o` is not `COLLINEAR`.

template<typename RatKernel , typename AlgKernel , typename NtTraits >

CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::Curve_2 | ( | const typename RatKernel::Point_2 & | p1,
|  | const typename RatKernel::Point_2 & | p2,
|  | const typename RatKernel::Point_2 & | p3
| ) |  |

constructs a circular arc going from `p1` (its source point) through `p2` to `p3` (its target point).

Note that all three points have rational coordinates. The orientation of the arc is determined automatically.

Precondition
  The three points are not collinear.

template<typename RatKernel , typename AlgKernel , typename NtTraits >

CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::Curve_2 | ( | const Rational & | r,
|  | const Rational & | s,
|  | const Rational & | t,
|  | const Rational & | u,
|  | const Rational & | v,
|  | const Rational & | w
| ) |  |

constructs a conic arc that corresponds to the full conic curve \( r x^2 + s y^2 + t x y + u x + v y + w = 0\).

Precondition
  As a conic arc must be bounded, the given curve must be an ellipse, that is \( 4 r s - t^2 > 0\).

template<typename RatKernel , typename AlgKernel , typename NtTraits >

CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::Curve_2 | ( | const Rational & | r,
|  | const Rational & | s,
|  | const Rational & | t,
|  | const Rational & | u,
|  | const Rational & | v,
|  | const Rational & | w,
|  | Orientation | o,
|  | const Point_2 & | ps,
|  | const Point_2 & | pt
| ) |  |

constructs a conic arc supported by the conic curve \( r x^2 + s y^2 + t x y + u x + v y + w = 0\), going in the given orientation `o` from the source point `ps` to its target point `pt`.

Precondition
  `ps` and `pt` both satisfy the equation of the supporting conic curve and define a bounded segment of this curve (e.g. in case of a hyperbolic arc, both point should be located on the same branch of the hyperbola).
  `o` is not `COLLINEAR` if the supporting conic is curves, and must be `COLLINEAR` if it is not curved (a line or a line-pair).

template<typename RatKernel , typename AlgKernel , typename NtTraits >

CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::Curve_2 | ( | const typename RatKernel::Point_2 & | p1,
|  | const typename RatKernel::Point_2 & | p2,
|  | const typename RatKernel::Point_2 & | p3,
|  | const typename RatKernel::Point_2 & | p4,
|  | const typename RatKernel::Point_2 & | p5
| ) |  |

constructs a conic arc going from `p1` (its source point) through `p2`, `p3` and `p4` (in this order) to `p5` (its target point).

Note that all five points have rational coordinates. The orientation of the arc is determined automatically.

Precondition
  No three points of the five are not collinear.
  The five points define a valid arc, in their given order.

template<typename RatKernel , typename AlgKernel , typename NtTraits >

CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::Curve_2 | ( | const Rational & | r,
|  | const Rational & | s,
|  | const Rational & | t,
|  | const Rational & | u,
|  | const Rational & | v,
|  | const Rational & | w,
|  | Orientation | o,
|  | const Point_2 & | app_ps,
|  | const Rational & | r1,
|  | const Rational & | s1,
|  | const Rational & | t1,
|  | const Rational & | u1,
|  | const Rational & | v1,
|  | const Rational & | w1,
|  | const Point_2 & | app_pt,
|  | const Rational & | r2,
|  | const Rational & | s2,
|  | const Rational & | t2,
|  | const Rational & | u2,
|  | const Rational & | v2,
|  | const Rational & | w2
| ) |  |

constructs a conic arc supported by the conic curve \( r x^2 + s y^2 + t x y + u x + v y + w = 0\), going in the given orientation `o` from its source point to its target Point.

In this case only some approximations of the endpoints (`app_ps` and `app_pt`, respectively) is available, and their exact locations are given implicitly, specified by the intersections of the supporting conic curve with \( r_1 x^2 + s_1 y^2 + t_1 x y + u_1 x + v_1 y + w_1 = 0\) and \( r_2 x^2 + s_2 y^2 + t_2 x y + u_2 x + v_2 y + w_2 = 0\), respectively.

Precondition
  The two auxiliary curves specifying the endpoints really intersect with the supporting conic curve, such that the arc endpoints define a bounded segment of the supporting curve (e.g. in case of a hyperbolic arc, both point should be located on the same branch of the hyperbola).
  `o` is not `COLLINEAR` if the supporting conic is curves, and must be `COLLINEAR` if it is not curved (a line or a line-pair).

## Member Function Documentation

template<typename RatKernel , typename AlgKernel , typename NtTraits >

bool CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::is_valid | ( | ) |  | const

indicates whether `a` is a valid conic arc.

As the precondition to some of the constructor listed above are quite complicated, their violation does not cause the program to abort. Instead, the constructed arc is invalid (a defaultly constructed arc is also invalid). It is however recommended to check that a constructed arc is valid before inserting it to an arrangement, as this operation will cause the program to abort.

template<typename RatKernel , typename AlgKernel , typename NtTraits >

bool CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::is_x_monotone | ( | ) |  | const

determines whether the arc is \( x\)-monotone, namely each vertical line intersects it at most once.

A vertical line segment is also considered (weakly) \( x\)-monotone.

template<typename RatKernel , typename AlgKernel , typename NtTraits >

bool CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::is_y_monotone | ( | ) |  | const

determines whether the arc is \( y\)-monotone, namely each horizontal line intersects it at most once.

A horizontal line segment is also considered (weakly) \( x\)-monotone.

template<typename RatKernel , typename AlgKernel , typename NtTraits >

void CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::set_source | ( | const Point_2 & | ps) |  |

sets a new source point for the conic arc.

Precondition
  `ps` lies on the supporting conic of `a`.

template<typename RatKernel , typename AlgKernel , typename NtTraits >

void CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::set_target | ( | const Point_2 & | pt) |  |

sets a new target point for the conic arc.

Precondition
  `pt` lies on the supporting conic of `a`.

template<typename RatKernel , typename AlgKernel , typename NtTraits >

const Point_2& CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::source | ( | ) |  | const

returns the source point of the arc.

Precondition
  `a` is not a full conic curve.

template<typename RatKernel , typename AlgKernel , typename NtTraits >

const Point_2& CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits >::Curve_2::target | ( | ) |  | const

returns the target point of the arc.

Precondition
  `a` is not a full conic curve.