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

CGAL 4.8 - 2D Arrangements

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CGAL::Arr_rational_function_traits_2< AlgebraicKernel_d_1 >::Point_2 Class Reference

`#include <CGAL/Arr_rational_function_traits_2.h>`

## Definition

Is Model Of:
  `ArrTraits::Point_2`

## Types
typedef AlgebraicKernel_d_1::Polynomial_1 | Polynomial_1
typedef AlgebraicKernel_d_1::Algebraic_real_1 | Algebraic_real_1
typedef AlgebraicKernel_d_1::Bound | Bound

## Operations
Polynomial_1 | numerator () const
| returns the numerator of the supporting rational function.
Polynomial_1 | denominator () const
| returns the denominator of the supporting rational function.
std::pair< double, double > | to_double () const
| returns double-approximations of the x- and y-coordinates.
Algebraic_real_1 | x () const
| returns the \( x\)-coordinate of the point.
Algebraic_real_1 | y () const
| obtains the y-coordinates of the point. More...
std::pair< Bound, Bound > | approximate_absolute_x (int a) const
| Computes a pair \( p\) approximating the \( x\)-coordinate with respect to the given absolute precision \( a\). More...
std::pair< Bound, Bound > | approximate_absolute_y (int a) const
| Computes a pair \( p\) approximating the \( y\)-coordinate with respect to the given absolute precision \( a\). More...
std::pair< Bound, Bound > | approximate_relative_x (int r) const
| Computes a pair \( p\) approximating the \( x\)-coordinate with respect to the given relative precision \( r\). More...
std::pair< Bound, Bound > | approximate_relative_y (int r) const
| Computes a pair \( p\) approximating the \( y\)-coordinate with respect to the given relative precision \( r\). More...

## Member Function Documentation

template<typename AlgebraicKernel_d_1 >

std::pair<Bound,Bound> CGAL::Arr_rational_function_traits_2< AlgebraicKernel_d_1 >::Point_2::approximate_absolute_x | ( | int | a) |  | const

Computes a pair \( p\) approximating the \( x\)-coordinate with respect to the given absolute precision \( a\).

Postcondition
  \( p.first \leq x \leq p.second \)
  \( p.second - p.first \leq2^{-a} \)

template<typename AlgebraicKernel_d_1 >

std::pair<Bound,Bound> CGAL::Arr_rational_function_traits_2< AlgebraicKernel_d_1 >::Point_2::approximate_absolute_y | ( | int | a) |  | const

Computes a pair \( p\) approximating the \( y\)-coordinate with respect to the given absolute precision \( a\).

Postcondition
  \( p.first \leq y \leq p.second \)
  \( p.second - p.first \leq2^{-a} \)

template<typename AlgebraicKernel_d_1 >

std::pair<Bound,Bound> CGAL::Arr_rational_function_traits_2< AlgebraicKernel_d_1 >::Point_2::approximate_relative_x | ( | int | r) |  | const

Computes a pair \( p\) approximating the \( x\)-coordinate with respect to the given relative precision \( r\).

Postcondition
  \( p.first \leq x \leq p.second \)
  \( p.second - p.first \leq2^{-r}|x| \)

template<typename AlgebraicKernel_d_1 >

std::pair<Bound,Bound> CGAL::Arr_rational_function_traits_2< AlgebraicKernel_d_1 >::Point_2::approximate_relative_y | ( | int | r) |  | const

Computes a pair \( p\) approximating the \( y\)-coordinate with respect to the given relative precision \( r\).

Postcondition
  \( p.first \leq y \leq p.second \)
  \( p.second - p.first \leq2^{-r}|y| \)

template<typename AlgebraicKernel_d_1 >

Algebraic_real_1 CGAL::Arr_rational_function_traits_2< AlgebraicKernel_d_1 >::Point_2::y | ( | ) |  | const

obtains the y-coordinates of the point.

Attention: As described above, points are not stored by their y-coordinate in `Algebraic_real_1` representation. In fact, this representation must be computed on demand, and might become quite costly for points defined by high-degree polynomials. Therefore, it is recommended to avoid calls to this function as much as possible.