\( \newcommand{\E}{\mathrm{E}} \) \( \newcommand{\A}{\mathrm{A}} \) \( \newcommand{\R}{\mathrm{R}} \) \( \newcommand{\N}{\mathrm{N}} \) \( \newcommand{\Q}{\mathrm{Q}} \) \( \newcommand{\Z}{\mathrm{Z}} \) \( \def\ccSum #1#2#3{ \sum_{#1}^{#2}{#3} } \def\ccProd #1#2#3{ \sum_{#1}^{#2}{#3} }\)
CGAL 4.13 - 2D Arrangements
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CGAL::Arr_conic_traits_2< RatKernel, AlgKernel, NtTraits > Class Template Reference
2D Arrangement Reference » Traits Classes

#include <CGAL/Arr_conic_traits_2.h>

Definition

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, motion planning for a disc robot, etc.). Circular arcs and line segment are simpler objects and can be dealt with more efficiently than arbitrary arcs. For these reasons, it is possible to construct conic arcs from segments and from circles. Using these constructors is highly recommended: It is more straightforward and also speeds up the arrangement construction. However, in case the set of input curves contain only circular arcs and line segments, it is recommended to use the Arr_circle_segment_2 class to achieve faster 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 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, 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 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.

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.

Is Model Of:

ArrangementTraits_2

ArrangementLandmarkTraits_2

ArrangementDirectionalXMonotoneTraits_2

Types

Examples:
Arrangement_on_surface_2/conic_multiplicities.cpp, and Arrangement_on_surface_2/conics.cpp.

Classes

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  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).