CGAL 5.0  2D Arrangements

#include <CGAL/Arr_segment_traits_2.h>
The traits class Arr_segment_traits_2
is a model of the ArrangementTraits_2
concept, which allows the construction and maintenance of arrangements of line segments.
It should be parameterized with a CGALkernel model that is templated in turn with a number type. To avoid numerical errors and robustness problems, the number type should support exact rational arithmetic  that is, the number type should support the arithmetic operations \( +\), \( \), \( \times\) and \( \div\) carried out without loss of precision.
For example, instantiating the traits template with kernels that support exact predicates and exact constructions, such as Exact_predicates_exact_constructions_kernel
, ensures the exact and robust operation of the application. While selecting an inexact kernel when developing a program usually leads to shorter running times, it causes robustness problems in most cases, and thus renders the program useless. The Exact_predicates_exact_constructions_kernel
type is a shortcut for a kernel that uses an exact number type, that is, Gmpq
(More precisely, Lazy_exact_nt<Gmpq>
), when Gmpq
is available. Indeed, this configuration achieves the shortest running times in many cases. If no type is provided for the kernel template parameter, then Exact_predicates_exact_constructions_kernel
will be used by default.
Arr_segment_traits_2
defines Kernel::Point_2
as its point type. However, it does not define Kernel::Segment_2
as its curve type, as one may expect. The reason is that the kernel segment is represented by its two endpoints only, while the traits class needs to store extra data with its segments, in order to efficiently operate on them. Nevertheless, the nested X_monotone_curve_2
and Curve_2
types (in this case both types refer to the same class, as every line segment is (weakly) \( x\)monotone) can however be converted to the type Kernel::Segment_2
.
Arr_segment_traits_2
achieves faster running times than the Arr_non_caching_segment_traits_2<Kernel>
traitsclass, when arrangements with relatively many intersection points are constructed. It also allows for working with less accurate, yet computationally efficient number types, such as Quotient<MP_Float>
, which represents floatingpoint numbers with an unbounded mantissa, but with a bounded exponent. Using this traits class is therefore highly recommended for almost all applications that rely on arrangements of line segments. On the other hand, Arr_segment_traits_2
uses more space and stores extra data with each segment, so constructing arrangements of huge sets of nonintersecting segments (or segments that intersect very sparsely) could be more efficient with the Arr_non_caching_segment_traits_2
traitsclass.
While Arr_segment_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_segment_traits_2
supports the merging of curves of opposite directions.