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CGAL 6.1 - 2D Arrangements
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CGAL 6.1 - 2D Arrangements
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#include <CGAL/Arr_conic_traits_2.h>
The class Arr_conic_traits_2 is a model of the ArrangementTraits_2 concept and can be used to construct and maintain arrangements of bounded segments of algebraic curves of degree \(2\) at most, also known as conic curves.
A general conic curve \(C\) is the locus of all points \((x,y)\) satisfying the equation: \(r x^2 + s y^2 + t x y + u x + v y + w = 0\), where:
If \(4 r s - t^2 > 0\), \( C\) is an ellipse. A special case occurs when \(r = s\) and \( t = 0\), when \( C\) becomes a circle.
If \(4 r s - t^2 < 0\), \( C\) is a hyperbola.
If \(4 r s - t^2 = 0\), \( C\) is a parabola. A degenerate case occurs when \(r = s = t = 0\), when \( C\) is a line.
A bounded conic arc is defined as either of the following:
A full ellipse (or a circle) \( C\).
The tuple \( \langle C, p_s, p_t, o \rangle\), where \( C\) is the supporting conic curve, with the arc endpoints being \( p_s\) and \(
p_t\) (the source and target points, respectively). The orientation \( o\) indicates whether we proceed from \( p_s\) to \( p_t\) in a clockwise or in a counterclockwise direction. Note that \( C\) may also correspond to a line or to pair of lines—in this case \( o\) may specify a COLLINEAR orientation.
A very useful subset of the set of conic arcs are line segments and circular arcs, as arrangements of circular arcs and line segments have some interesting applications (e.g., offsetting polygons and motion planning for a disc robot). Circular arcs and line segment are simpler objects and can be dealt with more efficiently than arbitrary arcs. Indeed, it is possible to construct conic arcs from segments and from circles. Using these constructors is highly recommended: It is more straightforward and also expedites the arrangement construction. However, in case the set of input curves contain only circular arcs and line segments, it is recommended using the Arr_circle_segment_2 class to achieve better running times.
In our representation, all conic coefficients (namely \(r, s, t, u, v, w\)) must be rational numbers. This guarantees that the coordinates of all arrangement vertices (in particular, those representing intersection points) are algebraic numbers of degree \(4\) (a real number \(\alpha\) is an algebraic number of degree \(d\) if there exist a polynomial \( p\) with integer coefficient of degree \(d\) such that \(p(\alpha) = 0\)). We therefore require separate representations of the curve coefficients and the point coordinates. The NtTraits should be substituted by a class that defines nested Integer, Rational, and Algebraic number types and supports various operations on them, yielding certified computation results (for example, it 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 instantiated with the NtTraits::Rational and NtTraits::Algebraic number types, respectively. It is recommended substituting the CORE_algebraic_number_traits class for the NtTraits parameter, and the Cartesian<NtTraits::Rational> and Cartesian<NtTraits::Algebraic> instances for 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.
The traits class inherits its point type from AlgKernel::Point_2, and defines a curve and \(x\)-monotone curve types, as detailed below.
While the Arr_conic_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_conic_traits_2 supports the merging of curves of opposite directions.
Types
Classes | |
| class | Approximate_2 |
| A functor that approximates a point and an \(x\)-monotone curve. More... | |
| class | Construct_bbox_2 |
| A functor that constructs a bounding box of a conic arc. More... | |
| class | Construct_curve_2 |
| A functor that constructs a conic arc. More... | |
| class | Construct_x_monotone_curve_2 |
| A functor that constructs an \(x\)-monotone conic arc. More... | |
| class | Curve_2 |
The Curve_2 class nested within the conic-arc traits can represent arbitrary conic arcs and support their construction in various ways. More... | |
| class | Point_2 |
The Point_2 class nested within the conic-arc traits is used to represent points. More... | |
| class | Trim_2 |
| A functor that trims a conic arc. More... | |
| class | X_monotone_curve_2 |
The X_monotone_curve_2 class nested within the conic-arc traits is used to represent \(x\)-monotone conic arcs. 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). | |
Auxiliary Functor definitions, used gor, e.g., the landmarks | |
point-location strategy and the drawing function. | |
| typedef double | Approximate_number_type |
| typedef CGAL::Cartesian< Approximate_number_type > | Approximate_kernel |
| typedef Approximate_kernel::Point_2 | Approximate_point_2 |
Accessing Functor Objects | |
| Construct_curve_2 | construct_curve_2_object () const |
Obtain a Construct_curve_2 functor. | |
| Construct_x_monotone_curve_2 | construct_x_monotone_curve_2_object () const |
Obtain a Construct_x_monotone_curve_2 functor. | |
| Construct_bbox_2 | construct_bbox_2_object () const |
Obtain a Bbox_2 functor. | |
| Trim_2 | trim_2_object () const |
Obtain a Trim_2 functor. | |
| Approximate_2 | approximate_2_object () const |
Obtain an Approximate_2 functor. | |
1.9.6