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CGAL 4.8 - 2D Arrangements

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CGAL::Arr_Bezier_curve_traits_2< RatKernel, AlgKernel, NtTraits > Class Template Reference
Traits Classes

`#include <CGAL/Arr_Bezier_curve_traits_2.h>`

## Definition

The traits class `Arr_Bezier_curve_traits_2` is a model of the `ArrangementTraits_2` concept that handles planar Bézier curves.

A planar Bézier curve \( B\) is a parametric curve defined by a sequence of control points \( p_0, \ldots, p_n\) as follows:

\begin{eqnarray*} B(t) = \left(X(t), Y(t)\right) = \ccSum{k=0}{n}{p_k \cdot \frac{n!}{k! (n-k)!} \cdot t^k (1-t)^{n-k}}\ . \end{eqnarray*}

where \( t \in [0, 1]\). The degree of the curve is therefore \( n\) - namely, \( X(t)\) and \( Y(t)\) are polynomials of degree \( n\). Bézier curves have numerous applications in computer graphics and solid modelling. They are used, for example, in free-form sketches and for defining the true-type fonts.

In our representation, we assume that the coordinates of all control points are rational numbers (namely they are given as objects of the `RatKernel::Point_2` type), so both \( X(t)\) and \( Y(t)\) are polynomials with rational coefficients. The intersection points between curves are however algebraic numbers, and their exact computation is time-consuming. The traits class therefore contains a layer of geometric filtering that performs all computation in an approximate manner whenever possible, and it resorts to exact computations only when the approximate computation fails to produce an unambiguous result.

We therefore require separate representations of the control points and the intersection points. The `NtTraits` should be instantiated with a class that defines nested `Integer`, `Rational` and `Algebraic` number types and supports various operations on them, yielding certified computation results (for example, in can convert rational numbers to algebraic numbers and can compute roots of polynomials with integer coefficients). The other template parameters, `RatKernel` and `AlgKernel` should be geometric kernels templated with the `NtTraits::Rational` and `NtTraits::Algebraic` number types, respectively. It is recommended to instantiate the `CORE_algebraic_number_traits` class as the `NtTraits` parameter, with `Cartesian<NtTraits::Rational>` and `Cartesian<NtTraits::Algebraic>` instantiating the two kernel types, respectively. The number types in this case are provided by the Core library, with its ability to exactly represent simple algebraic numbers.

While `Arr_Bezier_curve_traits_2` models the concept `ArrangementDirectionalXMonotoneTraits_2`, the implementation of the `Are_mergeable_2` operation does not enforce the input curves to have the same direction as a precondition. Moreover, `Arr_Bezier_curve_traits_2` supports the merging of curves of opposite directions.

Is Model Of:

  `ArrangementTraits_2`

  `ArrangementDirectionalXMonotoneTraits_2`

Examples:
  Arrangement_on_surface_2/Bezier_curves.cpp.

## Classes
class | Curve_2
| The `Curve_2` class nested within the Bézier traits class is used to represent a Bézier curve of arbitrary degree, which is defined by a sequence of rational control points. More...
class | Point_2
| The `Point_2` class nested within the Bézier traits class is used to represent: (i) an endpoint of a Bézier curve, (ii) a vertical tangency point of a curve, used to subdivide it into \( x\)-monotone subcurve, and (iii) an intersection point between two curves. More...
class | X_monotone_curve_2
| The `X_monotone_curve_2` class nested within the Bézier traits is used to represent \( x\)-monotone subcurves of Bézier curves. More...

## Types
typedef unspecified_type | Rational
| the `NtTraits::Rational` type (and also the `RatKernel::FT` type).
typedef unspecified_type | Algebraic
| the `NtTraits::Algebraic` type (and also the `AlgKernel::FT` type).