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// Copyright (c) 2005 Tel-Aviv University (Israel).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org); you may redistribute it under
// the terms of the Q Public License version 1.0.
// See the file LICENSE.QPL distributed with CGAL.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL$
// $Id$
//
//
// Author(s) : Ron Wein <wein@post.tau.ac.il>
#ifndef CGAL_ARR_CONIC_TRAITS_2_H
#define CGAL_ARR_CONIC_TRAITS_2_H
/*! \file
* The conic traits-class for the arrangement package.
*/
#include <CGAL/tags.h>
#include <CGAL/Arr_traits_2/Conic_arc_2.h>
#include <CGAL/Arr_traits_2/Conic_x_monotone_arc_2.h>
#include <CGAL/Arr_traits_2/Conic_point_2.h>
#include <fstream>
CGAL_BEGIN_NAMESPACE
/*!
* \class A traits class for maintaining an arrangement of conic arcs (bounded
* segments of algebraic curves of degree 2 at most).
*
* The class is templated with two parameters:
* Rat_kernel A kernel that provides the input objects or coefficients.
* Rat_kernel::FT should be an integral or a rational type.
* Alg_kernel A geometric kernel, where Alg_kernel::FT is the number type
* for the coordinates of arrangement vertices, which are algebraic
* numbers of degree up to 4 (preferably it is CORE::Expr).
* Nt_traits A traits class for performing various operations on the integer,
* rational and algebraic types.
*/
template <class Rat_kernel_, class Alg_kernel_, class Nt_traits_>
class Arr_conic_traits_2
{
public:
typedef Rat_kernel_ Rat_kernel;
typedef Alg_kernel_ Alg_kernel;
typedef Nt_traits_ Nt_traits;
typedef typename Rat_kernel::FT Rational;
typedef typename Rat_kernel::Point_2 Rat_point_2;
typedef typename Rat_kernel::Segment_2 Rat_segment_2;
typedef typename Rat_kernel::Line_2 Rat_line_2;
typedef typename Rat_kernel::Circle_2 Rat_circle_2;
typedef typename Alg_kernel::FT Algebraic;
typedef typename Nt_traits::Integer Integer;
typedef Arr_conic_traits_2<Rat_kernel, Alg_kernel, Nt_traits> Self;
// Category tags:
typedef Tag_true Has_left_category;
typedef Tag_true Has_merge_category;
// Traits objects:
typedef _Conic_arc_2<Rat_kernel, Alg_kernel, Nt_traits> Curve_2;
typedef _Conic_x_monotone_arc_2<Curve_2> X_monotone_curve_2;
typedef _Conic_point_2<Alg_kernel> Point_2;
private:
// Type definition for the intersection points mapping.
typedef typename X_monotone_curve_2::Conic_id Conic_id;
typedef typename X_monotone_curve_2::Intersection_point_2
Intersection_point_2;
typedef typename X_monotone_curve_2::Intersection_map Intersection_map;
Intersection_map inter_map; // Mapping conic pairs to their intersection
// points.
public:
/*!
* Default constructor.
*/
Arr_conic_traits_2 ()
{}
/*! Get the next conic index. */
static unsigned int get_index ()
{
static unsigned int index = 0;
return (++index);
}
/// \name Basic functor definitions.
//@{
class Compare_x_2
{
public:
/*!
* Compare the x-coordinates of two points.
* \param p1 The first point.
* \param p2 The second point.
* \return LARGER if x(p1) > x(p2);
* SMALLER if x(p1) < x(p2);
* EQUAL if x(p1) = x(p2).
*/
Comparison_result operator() (const Point_2 & p1, const Point_2 & p2) const
{
Alg_kernel ker;
return (ker.compare_x_2_object() (p1, p2));
}
};
/*! Get a Compare_x_2 functor object. */
Compare_x_2 compare_x_2_object () const
{
return Compare_x_2();
}
class Compare_xy_2
{
public:
/*!
* Compares two points lexigoraphically: by x, then by y.
* \param p1 The first point.
* \param p2 The second point.
* \return LARGER if x(p1) > x(p2), or if x(p1) = x(p2) and y(p1) > y(p2);
* SMALLER if x(p1) < x(p2), or if x(p1) = x(p2) and y(p1) < y(p2);
* EQUAL if the two points are equal.
*/
Comparison_result operator() (const Point_2& p1, const Point_2& p2) const
{
Alg_kernel ker;
return (ker.compare_xy_2_object() (p1, p2));
}
};
/*! Get a Compare_xy_2 functor object. */
Compare_xy_2 compare_xy_2_object () const
{
return Compare_xy_2();
}
class Construct_min_vertex_2
{
public:
/*!
* Get the left endpoint of the x-monotone curve (segment).
* \param cv The curve.
* \return The left endpoint.
*/
const Point_2& operator() (const X_monotone_curve_2 & cv) const
{
return (cv.left());
}
};
/*! Get a Construct_min_vertex_2 functor object. */
Construct_min_vertex_2 construct_min_vertex_2_object () const
{
return Construct_min_vertex_2();
}
class Construct_max_vertex_2
{
public:
/*!
* Get the right endpoint of the x-monotone curve (segment).
* \param cv The curve.
* \return The right endpoint.
*/
const Point_2& operator() (const X_monotone_curve_2 & cv) const
{
return (cv.right());
}
};
/*! Get a Construct_max_vertex_2 functor object. */
Construct_max_vertex_2 construct_max_vertex_2_object () const
{
return Construct_max_vertex_2();
}
class Is_vertical_2
{
public:
/*!
* Check whether the given x-monotone curve is a vertical segment.
* \param cv The curve.
* \return (true) if the curve is a vertical segment; (false) otherwise.
*/
bool operator() (const X_monotone_curve_2& cv) const
{
return (cv.is_vertical());
}
};
/*! Get an Is_vertical_2 functor object. */
Is_vertical_2 is_vertical_2_object () const
{
return Is_vertical_2();
}
class Compare_y_at_x_2
{
public:
/*!
* Return the location of the given point with respect to the input curve.
* \param cv The curve.
* \param p The point.
* \pre p is in the x-range of cv.
* \return SMALLER if y(p) < cv(x(p)), i.e. the point is below the curve;
* LARGER if y(p) > cv(x(p)), i.e. the point is above the curve;
* EQUAL if p lies on the curve.
*/
Comparison_result operator() (const Point_2 & p,
const X_monotone_curve_2 & cv) const
{
Alg_kernel ker;
if (cv.is_vertical())
{
// A special treatment for vertical segments:
// In case p has the same x c-ordinate of the vertical segment, compare
// it to the segment endpoints to determine its position.
Comparison_result res1 = ker.compare_y_2_object() (p, cv.left());
Comparison_result res2 = ker.compare_y_2_object() (p, cv.right());
if (res1 == res2)
return (res1);
else
return (EQUAL);
}
// Check whether the point is exactly on the curve.
if (cv.contains_point(p))
return (EQUAL);
// Get a point q on the x-monotone arc with the same x coordinate as p.
Comparison_result x_res;
Point_2 q;
if ((x_res = ker.compare_x_2_object() (p, cv.left())) == EQUAL)
{
q = cv.left();
}
else
{
CGAL_precondition (x_res != SMALLER);
if ((x_res = ker.compare_x_2_object() (p, cv.right())) == EQUAL)
{
q = cv.right();
}
else
{
CGAL_precondition (x_res != LARGER);
q = cv.get_point_at_x (p);
}
}
// Compare p with the a point of the curve with the same x coordinate.
return (ker.compare_y_2_object() (p, q));
}
};
/*! Get a Compare_y_at_x_2 functor object. */
Compare_y_at_x_2 compare_y_at_x_2_object () const
{
return Compare_y_at_x_2();
}
class Compare_y_at_x_left_2
{
public:
/*!
* Compares the y value of two x-monotone curves immediately to the left
* of their intersection point.
* \param cv1 The first curve.
* \param cv2 The second curve.
* \param p The intersection point.
* \pre The point p lies on both curves, and both of them must be also be
* defined (lexicographically) to its left.
* \return The relative position of cv1 with respect to cv2 immdiately to
* the left of p: SMALLER, LARGER or EQUAL.
*/
Comparison_result operator() (const X_monotone_curve_2& cv1,
const X_monotone_curve_2& cv2,
const Point_2& p) const
{
// Make sure that p lies on both curves, and that both are defined to its
// left (so their left endpoint is lexicographically smaller than p).
CGAL_precondition (cv1.contains_point (p) &&
cv2.contains_point (p));
CGAL_precondition_code (
Alg_kernel ker;
);
CGAL_precondition (ker.compare_xy_2_object() (p,
cv1.left()) == LARGER &&
ker.compare_xy_2_object() (p,
cv2.left()) == LARGER);
// If one of the curves is vertical, it is below the other one.
if (cv1.is_vertical())
{
if (cv2.is_vertical())
// Both are vertical:
return (EQUAL);
else
return (SMALLER);
}
else if (cv2.is_vertical())
{
return (LARGER);
}
// Compare the two curves immediately to the left of p:
return (cv1.compare_to_left (cv2, p));
}
};
/*! Get a Compare_y_at_x_left_2 functor object. */
Compare_y_at_x_left_2 compare_y_at_x_left_2_object () const
{
return Compare_y_at_x_left_2();
}
class Compare_y_at_x_right_2
{
public:
/*!
* Compares the y value of two x-monotone curves immediately to the right
* of their intersection point.
* \param cv1 The first curve.
* \param cv2 The second curve.
* \param p The intersection point.
* \pre The point p lies on both curves, and both of them must be also be
* defined (lexicographically) to its right.
* \return The relative position of cv1 with respect to cv2 immdiately to
* the right of p: SMALLER, LARGER or EQUAL.
*/
Comparison_result operator() (const X_monotone_curve_2& cv1,
const X_monotone_curve_2& cv2,
const Point_2& p) const
{
// Make sure that p lies on both curves, and that both are defined to its
// left (so their left endpoint is lexicographically smaller than p).
CGAL_precondition (cv1.contains_point (p) &&
cv2.contains_point (p));
CGAL_precondition_code (
Alg_kernel ker;
);
CGAL_precondition (ker.compare_xy_2_object() (p,
cv1.right()) == SMALLER &&
ker.compare_xy_2_object() (p,
cv2.right()) == SMALLER);
// If one of the curves is vertical, it is above the other one.
if (cv1.is_vertical())
{
if (cv2.is_vertical())
// Both are vertical:
return (EQUAL);
else
return (LARGER);
}
else if (cv2.is_vertical())
{
return (SMALLER);
}
// Compare the two curves immediately to the right of p:
return (cv1.compare_to_right (cv2, p));
}
};
/*! Get a Compare_y_at_x_right_2 functor object. */
Compare_y_at_x_right_2 compare_y_at_x_right_2_object () const
{
return Compare_y_at_x_right_2();
}
class Equal_2
{
public:
/*!
* Check if the two x-monotone curves are the same (have the same graph).
* \param cv1 The first curve.
* \param cv2 The second curve.
* \return (true) if the two curves are the same; (false) otherwise.
*/
bool operator() (const X_monotone_curve_2& cv1,
const X_monotone_curve_2& cv2) const
{
if (&cv1 == &cv2)
return (true);
return (cv1.equals (cv2));
}
/*!
* Check if the two points are the same.
* \param p1 The first point.
* \param p2 The second point.
* \return (true) if the two point are the same; (false) otherwise.
*/
bool operator() (const Point_2& p1, const Point_2& p2) const
{
if (&p1 == &p2)
return (true);
Alg_kernel ker;
return (ker.compare_xy_2_object() (p1, p2) == EQUAL);
}
};
/*! Get an Equal_2 functor object. */
Equal_2 equal_2_object () const
{
return Equal_2();
}
//@}
/// \name Functor definitions for supporting intersections.
//@{
class Make_x_monotone_2
{
typedef Arr_conic_traits_2 <Rat_kernel_, Alg_kernel_, Nt_traits_> Self;
public:
/*!
* Cut the given conic curve (or conic arc) into x-monotone subcurves
* and insert them to the given output iterator.
* \param cv The curve.
* \param oi The output iterator, whose value-type is Object. The returned
* objects are all wrappers X_monotone_curve_2 objects.
* \return The past-the-end iterator.
*/
template<class OutputIterator>
OutputIterator operator() (const Curve_2& cv, OutputIterator oi)
{
// Increment the serial number of the curve cv, which will serve as its
// unique identifier.
unsigned int index = Self::get_index();
Conic_id conic_id (index);
// Find the points of vertical tangency to cv and act accordingly.
typename Curve_2::Point_2 vtan_ps[2];
int n_vtan_ps;
n_vtan_ps = cv.vertical_tangency_points (vtan_ps);
if (n_vtan_ps == 0)
{
// In case the given curve is already x-monotone:
*oi = make_object (X_monotone_curve_2 (cv, conic_id));
++oi;
return (oi);
}
// Split the conic arc into x-monotone sub-curves.
if (cv.is_full_conic())
{
// Make sure we have two vertical tangency points.
CGAL_assertion(n_vtan_ps == 2);
// In case the curve is a full conic, split it into two x-monotone
// arcs, one going from ps[0] to ps[1], and the other from ps[1] to
// ps[0].
*oi = make_object (X_monotone_curve_2 (cv, vtan_ps[0], vtan_ps[1],
conic_id));
++oi;
*oi = make_object (X_monotone_curve_2 (cv, vtan_ps[1], vtan_ps[0],
conic_id));
++oi;
}
else
{
if (n_vtan_ps == 1)
{
// Split the arc into two x-monotone sub-curves: one going from the
// arc source to ps[0], and the other from ps[0] to the target.
*oi = make_object (X_monotone_curve_2 (cv, cv.source(), vtan_ps[0],
conic_id));
++oi;
*oi = make_object (X_monotone_curve_2 (cv, vtan_ps[0], cv.target(),
conic_id));
++oi;
}
else
{
CGAL_assertion (n_vtan_ps == 2);
// Identify the first point we encounter when going from cv's source
// to its target, and the second point we encounter. Note that the
// two endpoints must both be below the line connecting the two
// tangnecy points (or both lies above it).
int ind_first = 0;
int ind_second = 1;
Alg_kernel_ ker;
typename Alg_kernel_::Line_2 line =
ker.construct_line_2_object() (vtan_ps[0], vtan_ps[1]);
const Comparison_result start_pos =
ker.compare_y_at_x_2_object() (cv.source(), line);
const Comparison_result order_vpts =
ker.compare_x_2_object() (vtan_ps[0], vtan_ps[1]);
CGAL_assertion (start_pos != EQUAL &&
ker.compare_y_at_x_2_object() (cv.target(),
line) == start_pos);
CGAL_assertion (order_vpts != EQUAL);
if ((cv.orientation() == COUNTERCLOCKWISE &&
start_pos == order_vpts) ||
(cv.orientation() == CLOCKWISE &&
start_pos != order_vpts))
{
ind_first = 1;
ind_second = 0;
}
// Split the arc into three x-monotone sub-curves.
*oi = make_object (X_monotone_curve_2 (cv,
cv.source(),
vtan_ps[ind_first],
conic_id));
++oi;
*oi = make_object (X_monotone_curve_2 (cv,
vtan_ps[ind_first],
vtan_ps[ind_second],
conic_id));
++oi;
*oi = make_object (X_monotone_curve_2 (cv,
vtan_ps[ind_second],
cv.target(),
conic_id));
++oi;
}
}
return (oi);
}
};
/*! Get a Make_x_monotone_2 functor object. */
Make_x_monotone_2 make_x_monotone_2_object ()
{
return Make_x_monotone_2();
}
class Split_2
{
public:
/*!
* Split a given x-monotone curve at a given point into two sub-curves.
* \param cv The curve to split
* \param p The split point.
* \param c1 Output: The left resulting subcurve (p is its right endpoint).
* \param c2 Output: The right resulting subcurve (p is its left endpoint).
* \pre p lies on cv but is not one of its end-points.
*/
void operator() (const X_monotone_curve_2& cv, const Point_2 & p,
X_monotone_curve_2& c1, X_monotone_curve_2& c2) const
{
cv.split (p, c1, c2);
return;
}
};
/*! Get a Split_2 functor object. */
Split_2 split_2_object () const
{
return Split_2();
}
class Intersect_2
{
private:
Intersection_map& _inter_map; // The map of intersection points.
public:
/*! Constructor. */
Intersect_2 (Intersection_map& map) :
_inter_map (map)
{}
/*!
* Find the intersections of the two given curves and insert them to the
* given output iterator. As two segments may itersect only once, only a
* single will be contained in the iterator.
* \param cv1 The first curve.
* \param cv2 The second curve.
* \param oi The output iterator.
* \return The past-the-end iterator.
*/
template<class OutputIterator>
OutputIterator operator() (const X_monotone_curve_2& cv1,
const X_monotone_curve_2& cv2,
OutputIterator oi)
{
return (cv1.intersect (cv2, _inter_map, oi));
}
};
/*! Get an Intersect_2 functor object. */
Intersect_2 intersect_2_object ()
{
return (Intersect_2 (inter_map));
}
class Are_mergeable_2
{
public:
/*!
* Check whether it is possible to merge two given x-monotone curves.
* \param cv1 The first curve.
* \param cv2 The second curve.
* \return (true) if the two curves are mergeable - if they are supported
* by the same line and share a common endpoint; (false) otherwise.
*/
bool operator() (const X_monotone_curve_2& cv1,
const X_monotone_curve_2& cv2) const
{
return (cv1.can_merge_with (cv2));
}
};
/*! Get an Are_mergeable_2 functor object. */
Are_mergeable_2 are_mergeable_2_object () const
{
return Are_mergeable_2();
}
class Merge_2
{
public:
/*!
* Merge two given x-monotone curves into a single curve (segment).
* \param cv1 The first curve.
* \param cv2 The second curve.
* \param c Output: The merged curve.
* \pre The two curves are mergeable, that is they are supported by the
* same conic curve and share a common endpoint.
*/
void operator() (const X_monotone_curve_2& cv1,
const X_monotone_curve_2& cv2,
X_monotone_curve_2& c) const
{
c = cv1;
c.merge (cv2);
return;
}
};
/*! Get a Merge_2 functor object. */
Merge_2 merge_2_object () const
{
return Merge_2();
}
//@}
/// \name Functor definitions for the landmarks point-location strategy.
//@{
typedef double Approximate_number_type;
class Approximate_2
{
public:
/*!
* Return an approximation of a point coordinate.
* \param p The exact point.
* \param i The coordinate index (either 0 or 1).
* \pre i is either 0 or 1.
* \return An approximation of p's x-coordinate (if i == 0), or an
* approximation of p's y-coordinate (if i == 1).
*/
Approximate_number_type operator() (const Point_2& p,
int i) const
{
CGAL_precondition (i == 0 || i == 1);
if (i == 0)
return (CGAL::to_double(p.x()));
else
return (CGAL::to_double(p.y()));
}
};
/*! Get an Approximate_2 functor object. */
Approximate_2 approximate_2_object () const
{
return Approximate_2();
}
class Construct_x_monotone_curve_2
{
public:
/*!
* Return an x-monotone curve connecting the two given endpoints.
* \param p The first point.
* \param q The second point.
* \pre p and q must not be the same.
* \return A segment connecting p and q.
*/
X_monotone_curve_2 operator() (const Point_2& p,
const Point_2& q) const
{
return (X_monotone_curve_2 (p, q));
}
};
/*! Get a Construct_x_monotone_curve_2 functor object. */
Construct_x_monotone_curve_2 construct_x_monotone_curve_2_object () const
{
return Construct_x_monotone_curve_2();
}
//@}
/// \name Functor definitions for the Boolean set-operation traits.
//@{
class Compare_endpoints_xy_2
{
public:
/*!
* Compare the endpoints of an $x$-monotone curve lexicographically.
* (assuming the curve has a designated source and target points).
* \param cv The curve.
* \return SMALLER if the curve is directed right;
* LARGER if the curve is directed left.
*/
Comparison_result operator() (const X_monotone_curve_2& cv)
{
if (cv.is_directed_right())
return (SMALLER);
else
return (LARGER);
}
};
/*! Get a Compare_endpoints_xy_2 functor object. */
Compare_endpoints_xy_2 compare_endpoints_xy_2_object() const
{
return Compare_endpoints_xy_2();
}
class Construct_opposite_2
{
public:
/*!
* Construct an opposite x-monotone (with swapped source and target).
* \param cv The curve.
* \return The opposite curve.
*/
X_monotone_curve_2 operator() (const X_monotone_curve_2& cv)
{
return (cv.flip());
}
};
/*! Get a Construct_opposite_2 functor object. */
Construct_opposite_2 construct_opposite_2_object() const
{
return Construct_opposite_2();
}
//@}
};
CGAL_END_NAMESPACE
#endif
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