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

CGAL 4.8 - 2D Arrangements

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CGAL::Arr_polycurve_traits_2< GeometryTraits_2 > Class Template Reference
Traits Classes

`#include <CGAL/Arr_polycurve_traits_2.h>`

## Definition

Note: The `GeometryTraits_2` can comprise of Line_segments, Conic_arcs, Circular_arc, Bezier_curves or Linear_curves.

A portion or a part of any of the above mentioned geometric traits is called a segment.

The traits class `Arr_polycurve_traits_2` handles piecewise curves that are not necessarily linear, such as conic arcs, circular arcs, Bezier curves, or line segments. We call such a compound curve a polycurve. A polycurve is a chain of subcurves, where each two neighboring subcurves in the chain share a common endpoint; that is, the polycurve is continuous. Furthermore, the target of the \(i\)th segement of a polycurve has to coincide with the source of the \(i+1\)st segment; that is, the polycurve has to be well-oriented. Note that it is possible to construct general polycurves that are neither continuous nor well-oriented, as it is impossible to enforce this precondition (using the set of predicates required by the relevant concepts, see below). However, such polycurves cannot be used for the actual computation of arrangements. The traits class template exploits the functionality of the `GeometryTraits_2` template-parameter to handle the subcurves that compose the polycurve.

The type substituting the template parameter `GeometryTraits_2` when the template Arr_polycurve_traits_2 is instantiated must be a model of the concepts

- `ArrangementTraits_2` and
- `ArrangementDirectionalXMonotoneTraits_2`.

If, in addition, the GeometryTraits_2 models the concept `ArrangementApproximateTraits_2` then `Arr_polycurve_traits_2` models this concept as well. The same holds for the concept `ArrangementOpenBoundaryTraits_2`. If no type is provided, then `Arr_segment_traits_2` (instantiated with `Exact_predicates_exact_constructions_kernel` as the kernel) is used. Otherwise, `Arr_algebraic_segment_traits_2<Coefficient>`, `Arr_Bezier_curve_traits_2<RatKernel, AlgKernel, NtTraits>`, `Arr_circle_segment_traits_2<Kernel>`, `Arr_conic_traits_2<RatKernel, AlgKernel, NtTraits>`, `Arr_linear_traits_2<Kernel>`, `Arr_non_caching_segment_traits_2<Kernel>`, `Arr_segment_traits_2<Kernel>`, `Arr_rational_function_traits_2<AlgebraicKernel_d_1>`, or any other model of the concepts above can be used.

The number type used by the injected segment traits should support exact rational arithmetic (that is, the number type should support the arithmetic operations \( +\), \( -\), \( \times\) and \( \div\) carried out without loss of precision), in order to avoid robustness problems, although other inexact number types could be used at the user's own risk.

A polycurve that comprises \(n > 0\) segments has \( n+1 \) segment end-points, and they are represented as objects of type `GeometryTraits_2::Point_2`. Since the notion of a vertex is reserved to 0-dimensional elements of an arrangement, we use, in this context, the notion of points in order to refer to the vertices of a polycurve. For example, an arrangement induced by a single non-self intersecting polycurve has exactly two vertices regardless of the number of segment end-points. Finally, the types `Segment_2` and `X_monotone_segment_2` nested in `Arr_polycurve_traits_2` are nothing but `GeometryTraits_2::Curve_2` and `GeometryTraits_2::X_monotone_curve_2`, respectively.

A note on Backwards compatibility
In CGAL version 4.2 (and earlier) any object of the `X_monotone_curve_2` type nested in `Arr_polycurve_traits_2` which in that version was called `Arr_polyline_tratis_2` maintained a direction invariant; namely, its vertices were ordered in an ascending lexicographical \((xy)\)-order. This restriction is no longer imposed and `X_monotone_curve_2` can be now directed either from right-to-left or left-to-right. If you wish to maintain a left-to-right orientations of the \(x\)-monotone polycurve, set the macro `CGAL_ALWAYS_LEFT_TO_RIGHT` to 1 before any CGAL header is included.

Is Model Of:

  `ArrangementTraits_2`

  `ArrangementDirectionalXMonotoneTraits_2`

  `ArrangementApproximateTraits_2` (if the type that substitutes the template parameter `GeometryTraits_2` models the concept as well)

See Also
  `Arr_algebraic_segment_traits_2<Coefficient>`
  `Arr_Bezier_curve_traits_2<RatKernel,AlgKernel,NtTraits>`
  `Arr_circle_segment_traits_2<Kernel>`
  `Arr_conic_traits_2<RatKernel, AlgKernel, NtTraits>`
  `Arr_linear_traits_2<Kernel>`
  `Arr_non_caching_segment_traits_2<Kernel>`
  `Arr_segment_traits_2<Kernel>`
  `Arr_rational_function_traits_2<AlgebraicKernel_d_1>`
  `CGAL_ALWAYS_LEFT_TO_RIGHT`

## Classes
class | Construct_curve_2
| Construction functor of a general (not necessarily \(x\)-monotone) polycurve. More...
class | Construct_x_monotone_curve_2
| Construction functor of \(x\)-monotone polycurve. More...
class | Curve_2
| The `Curve_2` type nested in the `Arr_polycurve_traits_2` represents general continuous piecewise-linear segments (a polycurve can be self-intersecting) and support their construction from range of segments. More...
class | Make_x_monotone_2
| Subdivide the given segment into x-monotone sub-segments and insert them into the given output iterator. More...
class | Number_of_points_2
| Function object which returns the number of segment end-points of a polycurve. More...
class | Push_back_2
| Functor to augment a polycurve by either adding a vertex or a segment at the back. More...
class | Push_front_2
| Functor to augment a polycurve by either adding a vertex or a segment at the front. More...
class | Trim_2
class | X_monotone_curve_2
| The `X_monotone_curve_2` class nested within the polycurve traits is used to represent \( x\)-monotone piecewise linear segments. More...

## Types
typedef GeometryTraits_2::Point_2 | Point_2
typedef GeometryTraits_2::Curve_2 | Segment_2
typedef GeometryTraits_2::X_monotone_curve_2 | X_monotone_segment_2

## Accessing Functor Objects
Construct_curve_2 | construct_curve_2_object () const
Construct_x_monotone_curve_2 | construct_x_monotone_curve_2_object () const
Number_of_points_2 | number_of_points_2_object () const
Push_back_2 | push_back_2_object () const
Push_front_2 | push_front_2_object () const
Make_x_monotone_2 | make_x_monotone_2_object () const