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Integrate Instant-Meshes 1. Drop windows 32bit support Add windows 64bit support. 2. Remove Script(quickjs) support The current integrated quickjs is a very old version which doesn’t support windows 64bit system. In the future, (TODO)WebAssembly should be integrate as plugin system. 3. Integrate Instant-Meshes Instant-Meshes take less than 1 second on all three platforms. 4. Remove QuadriFlow Current implementation of QuadriFlow is too slow (take about 5 seconds on the example model) to do realtime remeshing.
2020-01-04 10:29:10 +00:00
// Copyright (c) 2005 Tel-Aviv University (Israel).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
// You can redistribute it and/or modify it under the terms of the GNU
// General Public License as published by the Free Software Foundation,
// either version 3 of the License, or (at your option) any later version.
//
// 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.
//
// $URL$
// $Id$
// SPDX-License-Identifier: GPL-3.0+
//
// Author(s) : Michal Meyerovitch <gorgymic@post.tau.ac.il>
// Baruch Zukerman <baruchzu@post.tau.ac.il>
/*! \file CGAL/Envelope_triangles_traits_3.h
* \brief Model for CGAL's EnvelopeTraits_3 concept.
* \endlink
*/
#ifndef CGAL_ENV_TRIANGLE_TRAITS_3_H
#define CGAL_ENV_TRIANGLE_TRAITS_3_H
#include <CGAL/license/Envelope_3.h>
#include <CGAL/Object.h>
#include <CGAL/enum.h>
#include <CGAL/Bbox_3.h>
#include <CGAL/Arr_segment_traits_2.h>
#include <CGAL/Envelope_3/Envelope_base.h>
#include <vector>
namespace CGAL {
template <class Kernel_> class Env_triangle_3;
// this traits class supports both triagles and segments in 3d
template <class Kernel_>
class Env_triangle_traits_3 : public Arr_segment_traits_2<Kernel_>
{
public:
typedef Arr_segment_traits_2<Kernel_> Traits_2;
typedef typename Traits_2::Point_2 Point_2;
typedef typename Traits_2::X_monotone_curve_2 X_monotone_curve_2;
typedef typename Traits_2::Multiplicity Multiplicity;
typedef Kernel_ Kernel;
typedef Env_triangle_traits_3<Kernel> Self;
typedef typename Kernel::Point_3 Point_3;
/*!
* \class Representation of a 3d triangle with cached data.
*/
class _Triangle_cached_3
{
public:
typedef typename Kernel::Plane_3 Plane_3;
typedef typename Kernel::Triangle_3 Triangle_3;
typedef typename Kernel::Point_3 Point_3;
typedef typename Kernel::Segment_3 Segment_3;
protected:
Plane_3 pl; // The plane that supports the triangle.
Point_3 vertices[3]; // The vertices of the triangle.
bool is_vert; // Is this a vertical triangle (or a segment).
bool is_seg; // Is this a segment.
public:
/*!
* Default constructor.
*/
_Triangle_cached_3() :
is_vert(false),
is_seg(false)
{}
/*!
* Constructor from a non-degenerate triangle.
* \param tri The triangle.
* \pre The triangle is not degenerate.
*/
_Triangle_cached_3(const Triangle_3 & tri)
{
Kernel kernel;
CGAL_assertion(!kernel.is_degenerate_3_object()(tri));
typename Kernel::Construct_vertex_3
construct_vertex = kernel.construct_vertex_3_object();
vertices[0] = construct_vertex(tri, 0);
vertices[1] = construct_vertex(tri, 1);
vertices[2] = construct_vertex(tri, 2);
pl = kernel.construct_plane_3_object()(vertices[0],
vertices[1], vertices[2]);
Self self;
is_vert = kernel.collinear_2_object()(self.project(vertices[0]),
self.project(vertices[1]),
self.project(vertices[2]));
is_seg = false;
}
/*!
* Construct a triangle from three non-collinear end-points.
* \param p1 The first point.
* \param p2 The second point.
* \param p3 The third point.
* \pre The 3 endpoints are not the collinear.
*/
_Triangle_cached_3(const Point_3 &p1, const Point_3 &p2,
const Point_3 &p3)
{
Kernel kernel;
CGAL_assertion(!kernel.collinear_3_object()(p1, p2, p3));
vertices[0] = p1;
vertices[1] = p2;
vertices[2] = p3;
pl = kernel.construct_plane_3_object()(vertices[0],
vertices[1],
vertices[2]);
Self self;
is_vert = kernel.collinear_2_object()(self.project(vertices[0]),
self.project(vertices[1]),
self.project(vertices[2]));
is_seg = false;
}
/*!
* Construct a triangle from 3 end-points on a supporting plane.
* \param supp_plane The supporting plane.
* \param p1 The first point.
* \param p2 The second point.
* \param p3 The third point.
* \pre The 3 endpoints are not the collinear and all lie on the given
* plane.
*/
_Triangle_cached_3(const Plane_3& supp_plane,
const Point_3 &p1,
const Point_3 &p2,
const Point_3 &p3) :
pl(supp_plane)
{
Kernel kernel;
CGAL_precondition(kernel.has_on_3_object() (pl, p1) &&
kernel.has_on_3_object() (pl, p2) &&
kernel.has_on_3_object() (pl, p3));
CGAL_precondition(!kernel.collinear_3_object()(p1, p2, p3));
vertices[0] = p1;
vertices[1] = p2;
vertices[2] = p3;
Self self;
is_vert = kernel.collinear_2_object()(self.project(vertices[0]),
self.project(vertices[1]),
self.project(vertices[2]));
is_seg = false;
}
/*!
* Constructor from a segment.
* \param seg The segment.
* \pre The segment is not degenerate.
*/
_Triangle_cached_3(const Segment_3 & seg)
{
Kernel kernel;
CGAL_assertion(!kernel.is_degenerate_3_object()(seg));
typename Kernel::Construct_vertex_3
construct_vertex = kernel.construct_vertex_3_object();
vertices[0] = construct_vertex(seg, 0);
vertices[1] = construct_vertex(seg, 1);
vertices[2] = vertices[1];
is_vert = true;
is_seg = true;
// construct a vertical plane through the segment
Point_3 tmp(vertices[0].x(), vertices[0].y(), vertices[0].z()-1);
pl = kernel.construct_plane_3_object()(vertices[0],
vertices[1], tmp);
}
/*!
* Constructor from two points.
* \param p1 The first point.
* \param p2 The second point.
* \param seg The segment.
* \pre The segment between the points is not degenerate.
*/
_Triangle_cached_3(const Point_3 &p1, const Point_3 &p2)
{
Kernel kernel;
CGAL_assertion(!kernel.equal_3_object()(p1, p2));
vertices[0] = p1;
vertices[1] = p2;
vertices[2] = p2;
is_vert = true;
is_seg = true;
// construct a vertical plane through the segment
Point_3 tmp(vertices[0].x(), vertices[0].y(), vertices[0].z()-1);
pl = kernel.construct_plane_3_object()(vertices[0],
vertices[1], tmp);
}
/*!
* Assignment operator.
* \param tri the source triangle to copy from
*/
const _Triangle_cached_3& operator=(const Triangle_3 &tri)
{
Kernel kernel;
CGAL_assertion(!kernel.is_degenerate_3_object()(tri));
typename Kernel_::Construct_vertex_3
construct_vertex = kernel.construct_vertex_3_object();
vertices[0] = construct_vertex(tri, 0);
vertices[1] = construct_vertex(tri, 1);
vertices[2] = construct_vertex(tri, 2);
pl = kernel.construct_plane_3_object()(vertices[0],
vertices[1], vertices[2]);
Self self;
is_vert = kernel.collinear_2_object()(self.project(vertices[0]),
self.project(vertices[1]),
self.project(vertices[2]));
is_seg = false;
return (*this);
}
/*!
* Get the ith endpoint.
*/
const Point_3& vertex(unsigned int i) const
{
return vertices[i%3];
}
/*!
* Get the supporting plane.
*/
const Plane_3& plane() const
{
return (pl);
}
/*!
* Check if the triangel is vertical.
*/
bool is_vertical() const
{
return (is_vert);
}
/*!
* Check if the surface is a segment.
*/
bool is_segment() const
{
return (is_seg);
}
/*!
* Check if the surface is xy-monotone (false, if it is a vertical
* triangle)
*/
bool is_xy_monotone() const
{
return (!is_vertical() || is_segment());
}
};
public:
// types for EnvelopeTraits_3 concept
//! type of xy-monotone surfaces
typedef Env_triangle_3<Kernel> Xy_monotone_surface_3;
//! type of surfaces
typedef Xy_monotone_surface_3 Surface_3;
// we have a collision between the Kernel's Intersect_2 and the one
// from the segment traits
typedef typename Traits_2::Intersect_2 Intersect_2;
protected:
typedef typename Kernel::FT FT;
typedef typename Kernel::Triangle_2 Triangle_2;
typedef typename Kernel::Segment_2 Segment_2;
typedef typename Kernel::Segment_3 Segment_3;
typedef typename Kernel::Triangle_3 Triangle_3;
typedef typename Kernel::Plane_3 Plane_3;
typedef typename Kernel::Assign_2 Assign_2;
typedef typename Kernel::Construct_vertex_2 Construct_vertex_2;
typedef typename Kernel::Assign_3 Assign_3;
typedef typename Kernel::Intersect_3 Intersect_3;
typedef typename Kernel::Construct_vertex_3 Construct_vertex_3;
typedef typename Kernel::Line_2 Line_2;
typedef typename Kernel::Direction_2 Direction_2;
typedef typename Kernel::Line_3 Line_3;
typedef typename Kernel::Direction_3 Direction_3;
typedef std::pair<X_monotone_curve_2,
Multiplicity> Intersection_curve;
public:
/***************************************************************************/
// EnvelopeTraits_3 functors
/***************************************************************************/
/*!\brief
* Subdivide the given surface into envelope relevant xy-monotone
* parts, and insert them into the output iterator.
*
* The iterator value-type is Xy_monotone_surface_3
*/
class Make_xy_monotone_3
{
protected:
const Self * parent;
public:
Make_xy_monotone_3(const Self * p) : parent(p)
{}
// create xy-monotone surfaces from a general surface
// return a past-the-end iterator
template <class OutputIterator>
OutputIterator operator()(const Surface_3& s,
bool is_lower,
OutputIterator o) const
{
m_is_lower = is_lower;
// a non-vertical triangle is already xy-monotone
if (!s.is_vertical())
*o++ = s;
else
{
// split a vertical triangle into one or two segments
const Point_3 &a1 = s.vertex(0),
a2 = s.vertex(1),
a3 = s.vertex(2);
Point_2 b1 = parent->project(a1),
b2 = parent->project(a2),
b3 = parent->project(a3);
Kernel k;
if (k.collinear_are_ordered_along_line_2_object()(b1, b2, b3))
{
if (k.equal_2_object()(b1, b2))
// only one segment in the output - the vertical does not count
*o++ = Xy_monotone_surface_3(find_envelope_point(a1, a2), a3);
else if (k.equal_2_object()(b2, b3))
*o++ = Xy_monotone_surface_3(a1, find_envelope_point(a2, a3));
else
// check whether two or one segments appear on the envelope
return find_envelope_segments(a1, a2, a3, s.plane(), o);
}
else if (k.collinear_are_ordered_along_line_2_object()(b1, b3, b2))
{
if (k.equal_2_object()(b1, b3))
// only one segment in the output
*o++ = Xy_monotone_surface_3(find_envelope_point(a1, a3), a2);
else
// check whether two or one segments appear on the envelope
return find_envelope_segments(a1, a3, a2, s.plane(), o);
}
else
{
// check whether two or one segments appear on the envelope
return find_envelope_segments(a2, a1, a3, s.plane(), o);
}
}
return o;
}
protected:
// find the envelope point among the two points with same xy coordinates
const Point_3& find_envelope_point (const Point_3& p1,
const Point_3& p2) const
{
CGAL_precondition(p1.x() == p2.x() && p1.y() == p2.y());
Kernel k;
Comparison_result cr = k.compare_z_3_object()(p1, p2);
CGAL_assertion(cr != EQUAL);
if ((m_is_lower && cr == SMALLER) ||
(!m_is_lower && cr == LARGER))
return p1;
else
return p2;
}
// get the three triangle coordinates (ordered along 2d-line) and find
// which segment(s) is(are) the envelope of this triangle
// "plane" is the vertical plane on which the triangle lies
template <class OutputIterator>
OutputIterator find_envelope_segments(const Point_3& p1,
const Point_3& p2,
const Point_3& p3,
const Plane_3& plane,
OutputIterator o) const
{
// our vertical plane is a*x + b*y + d = 0
FT a = plane.a(), b = plane.b();
CGAL_precondition(plane.c() == 0);
// if the plane was parallel to the yz-plane (i.e x = const),
// then it was enough to use the y,z coordinates as in the 2-dimensional
// case, to find whether a 2d point lies below/above/on a line
// this test is simply computing the sign of:
// (1) [(y3 - y1)(z2 - z1) - (z3 - z1)(y2 - y1)] * sign(y3 - y1)
// abd comparing to 0, where pi = (xi, yi, zi), and p2 is compared to the
// line formed by p1 and p3 (in the direction p1 -> p3)
//
// for general vertical plane, we change (x, y) coordinates to (v, w),
// (keeping the z-coordinate as is)
// so the plane is parallel to the wz-plane in the new coordinates
// (i.e v = const).
//
// ( v ) = A ( x ) where A = ( a b )
// w y -b a
//
// so v = a*x + b*y
// w = -b*x + a*y
//
// Putting the new points coordinates in equation (1) we get:
// (2) (w3 - w1)(z2 - z1) - (z3 - z1)(w2 - w1) =
// (-b*x3 + a*y3 + b*x1 - a*y1)(z2 - z1) -
// (z3 - z1)(-b*x2 + a*y2 + b*x1 - a*y1)
//
FT w1 = a*p1.y() - b*p1.x(),
w2 = a*p2.y() - b*p2.x(),
w3 = a*p3.y() - b*p3.x();
Sign s1 = CGAL::sign((w3 - w1)*(p2.z() - p1.z()) -
(p3.z() - p1.z())*(w2 - w1));
// the points should not be collinear
CGAL_assertion(s1 != 0);
// should also take care for the original and trasformed direction of
// the segment
Sign s2 = CGAL_NTS sign(w3 - w1);
Sign s = CGAL_NTS sign(int(s1 * s2));
bool use_one_segment = true;
if ((m_is_lower && s == NEGATIVE) ||
(!m_is_lower && s == POSITIVE))
use_one_segment = false;
if (use_one_segment)
{
*o++ = Xy_monotone_surface_3(p1, p3);
}
else
{
*o++ = Xy_monotone_surface_3(p1, p2);
*o++ = Xy_monotone_surface_3(p2, p3);
}
return o;
}
mutable bool m_is_lower;
};
/*! Get a Make_xy_monotone_3 functor object. */
Make_xy_monotone_3
make_xy_monotone_3_object() const
{
return Make_xy_monotone_3(this);
}
/*!\brief
* Insert all 2D curves, which form the boundary of the vertical
* projection of the surface onto the xy-plane, into the output iterator.
* The iterator value-type is X_monotone_curve_2.
*/
class Construct_projected_boundary_2
{
protected:
const Self *parent;
public:
Construct_projected_boundary_2(const Self* p)
: parent(p)
{}
// insert into the OutputIterator all the (2d) curves of the boundary of
// the vertical projection of the surface on the xy-plane
// the OutputIterator value type is X_monotone_curve_2
template <class OutputIterator>
OutputIterator operator()(const Xy_monotone_surface_3& s,
OutputIterator o) const
{
// the input xy-monotone surface should be either non-vertical or
// a segment
CGAL_assertion(s.is_xy_monotone());
if (!s.is_vertical())
{
// the projection is a triangle
const Point_3 &a1 = s.vertex(0),
a2 = s.vertex(1),
a3 = s.vertex(2);
Point_2 b1 = parent->project(a1),
b2 = parent->project(a2),
b3 = parent->project(a3);
Kernel k;
X_monotone_curve_2 A(b1, b2);
X_monotone_curve_2 B(b2, b3);
X_monotone_curve_2 C(b3, b1);
const Line_2& l1 =
(A.is_directed_right()) ? A.line() : A.line().opposite();
const Line_2& l2 =
(B.is_directed_right()) ? B.line() : B.line().opposite();
const Line_2& l3 =
(C.is_directed_right()) ? C.line() : C.line().opposite();
Oriented_side s1 = k.oriented_side_2_object()(l1, b3);
Oriented_side s2 = k.oriented_side_2_object()(l2, b1);
Oriented_side s3 = k.oriented_side_2_object()(l3, b2);
CGAL_assertion(s1 != ON_ORIENTED_BOUNDARY &&
s2 != ON_ORIENTED_BOUNDARY &&
s3 != ON_ORIENTED_BOUNDARY);
*o++ = make_object(std::make_pair(A, s1));
*o++ = make_object(std::make_pair(B, s2));
*o++ = make_object(std::make_pair(C, s3));
}
else
{
// s is a segment, and so is its projection
// s shouldn't be a z-vertical segment
const Point_3 &a1 = s.vertex(0),
a2 = s.vertex(1);
Point_2 b1 = parent->project(a1),
b2 = parent->project(a2);
CGAL_assertion(b1 != b2);
*o++ = make_object(std::make_pair(X_monotone_curve_2(b1, b2),
ON_ORIENTED_BOUNDARY));
}
return o;
}
};
/*! Get a Construct_projected_boundary_curves_2 functor object. */
Construct_projected_boundary_2
construct_projected_boundary_2_object() const
{
return Construct_projected_boundary_2(this);
}
/*!\brief
* Insert all the 2D projections (onto the xy-plane) of the
* intersection objects between s1 and s2 into the output iterator.
*
* The iterator value-type is Object. An Object may be:
* 1. A pair<X_monotone_curve_2,Intersection_type>, where the intersection
* type is an enumeration that can take the values
* {Transversal, Tangency, Unknown}.
* 2. A Point_2 instance (in degenerate cases).
*/
class Construct_projected_intersections_2
{
protected:
const Self *parent;
public:
Construct_projected_intersections_2(const Self* p)
: parent(p)
{}
// insert into OutputIterator all the (2d) projections on the xy plane of
// the intersection objects between the 2 surfaces
// the data type of OutputIterator is Object
template <class OutputIterator>
OutputIterator operator()(const Xy_monotone_surface_3& s1,
const Xy_monotone_surface_3& s2,
OutputIterator o) const
{
CGAL_assertion(s1.is_xy_monotone() && s2.is_xy_monotone());
Kernel k;
if (!parent->do_intersect(s1, s2))
{
return o;
}
Object inter_obj = parent->intersection(s1,s2);
if (inter_obj.is_empty())
{
return o;
}
Point_3 point;
Segment_3 curve;
if (k.assign_3_object()(point, inter_obj))
*o++ = make_object(parent->project(point));
else
{
CGAL_assertion_code(bool b = )
k.assign_3_object()(curve, inter_obj);
CGAL_assertion(b);
Segment_2 proj_seg = parent->project(curve);
if (! k.is_degenerate_2_object() (proj_seg))
{
Intersection_curve inter_cv (proj_seg, 1);
*o++ = make_object(inter_cv);
}
else
{
const Point_2& p = k.construct_point_on_2_object() (proj_seg, 0);
*o++ = make_object(p);
}
}
return o;
}
};
/*! Get a Construct_projected_intersections_2 functor object. */
Construct_projected_intersections_2
construct_projected_intersections_2_object() const
{
return Construct_projected_intersections_2(this);
}
/*!\brief
* Check if the surface s1 is closer/equally distanced/farther
* from the envelope with respect to s2 at the xy-coordinates of p/c.
*/
class Compare_z_at_xy_3
{
protected:
const Self *parent;
public:
Compare_z_at_xy_3(const Self* p)
: parent(p)
{}
// check which of the surfaces is closer to the envelope at the xy
// coordinates of point
// (i.e. lower if computing the lower envelope, or upper if computing
// the upper envelope)
// precondition: the surfaces are defined in point
Comparison_result operator()(const Point_2& p,
const Xy_monotone_surface_3& surf1,
const Xy_monotone_surface_3& surf2) const
{
// we compute the points on the planes, and then compare their z
// coordinates
const Plane_3& plane1 = surf1.plane();
const Plane_3& plane2 = surf2.plane();
// if the 2 triangles have the same supporting plane, and they are not
// vertical, then they have the same z coordinate over this point
if ((plane1 == plane2 || plane1 == plane2.opposite()) &&
!surf1.is_vertical())
{
return EQUAL;
}
Kernel k;
// Compute the intersetion between the vertical line and the given
// surfaces
Point_3 ip1 = parent->envelope_point_of_surface(p, surf1);
Point_3 ip2 = parent->envelope_point_of_surface(p, surf2);
return k.compare_z_3_object()(ip1, ip2);
}
// check which of the surfaces is closer to the envelope at the xy
// coordinates of cv
// (i.e. lower if computing the lower envelope, or upper if computing the
// upper envelope)
// precondition: the surfaces are defined in all points of cv,
// and the answer is the same for each of these points
Comparison_result operator()(const X_monotone_curve_2& cv,
const Xy_monotone_surface_3& surf1,
const Xy_monotone_surface_3& surf2) const
{
// first try the endpoints, if cannot be sure, use the mid point
Comparison_result res;
res = parent->compare_z_at_xy_3_object()(cv.left(), surf1, surf2);
if (res == EQUAL)
{
res = parent->compare_z_at_xy_3_object()(cv.right(), surf1, surf2);
if (res == EQUAL)
{
Point_2 mid = parent->construct_middle_point(cv);
res = parent->compare_z_at_xy_3_object()(mid, surf1, surf2);
}
}
return res;
}
};
/*! Get a Compare_z_at_xy_3 functor object. */
Compare_z_at_xy_3
compare_z_at_xy_3_object() const
{
return Compare_z_at_xy_3(this);
}
/*!\brief
* Check if the surface s1 is closer/equally distanced/farther
* from the envelope with
* respect to s2 immediately above the curve c.
*/
class Compare_z_at_xy_above_3
{
protected:
const Self *parent;
public:
Compare_z_at_xy_above_3(const Self* p)
: parent(p)
{}
// check which of the surfaces is closer to the envelope on the points
// above the curve cv
// (i.e. lower if computing the lower envelope, or upper if computing the
// upper envelope)
// precondition: the surfaces are defined above cv (to the left of cv,
// if cv is directed from min point to max point)
// the choise between surf1 and surf2 for the envelope is
// the same for every point in the infinitesimal region
// above cv
// the surfaces are EQUAL over the curve cv
Comparison_result
operator()(const X_monotone_curve_2& cv,
const Xy_monotone_surface_3& surf1,
const Xy_monotone_surface_3& surf2) const
{
// a vertical surface cannot be defined in the infinitesimal region above
// a curve
CGAL_precondition(!surf1.is_vertical());
CGAL_precondition(!surf2.is_vertical());
CGAL_precondition(parent->compare_z_at_xy_3_object()
(cv, surf1, surf2) == EQUAL);
CGAL_precondition(parent->compare_z_at_xy_3_object()
(cv.source(), surf1, surf2) == EQUAL);
CGAL_precondition(parent->compare_z_at_xy_3_object()
(cv.target(), surf1, surf2) == EQUAL);
if (parent->do_overlap(surf1, surf2))
{
return EQUAL;
}
// now we must have 2 different non-vertical planes:
// plane1: a1*x + b1*y + c1*z + d1 = 0 , c1 != 0
// plane2: a2*x + b2*y + c2*z + d2 = 0 , c2 != 0
const Plane_3& plane1 = surf1.plane();
const Plane_3& plane2 = surf2.plane();
FT a1 = plane1.a(), b1 = plane1.b(), c1 = plane1.c();
FT a2 = plane2.a(), b2 = plane2.b(), c2 = plane2.c();
// our line is a3*x + b3*y + c3 = 0
// it is assumed that the planes intersect over this line
const Line_2& line = cv.line();
FT a3 = line.a(), b3 = line.b(), c3 = line.c();
// if the line was parallel to the y-axis (i.e x = const),
// then it was enough to compare dz/dx of both planes
// for general line, we change coordinates to (v, w), preserving
// orientation, so the line is the w-axis in the new coordinates
// (i.e v = const).
//
// ( v ) = A ( x ) where A = ( a3 b3 )
// w y -b3 a3
//
// so v = a3*x + b3*y
// w = -b3*x + a3*y
// preserving orientation since detA = a3^2 +b3^2 > 0
//
// We compute the planes equations in the new coordinates
// and compare dz/dv
//
// ( x ) = A^(-1) ( v ) where A^(-1) = ( a3 -b3 ) * detA^(-1)
// y w b3 a3
// so x = (a3*v - b3*w)*(1/detA)
// y = (b3*v + a3*w)*(1/detA)
// plane1 ==> (a1a3 + b1b3)v + (b1a3 - a1b3)w + (c1z + d1)*detA = 0
// plane2 ==> (a2a3 + b2b3)v + (b2a3 - a2b3)w + (c2z + d2)*detA = 0
//
// dz/dv(1) = (-a1a3 - b1b3) / c1*detA
// dz/dv(2) = (-a2a3 - b2b3) / c2*detA
// since detA>0 we can omit it.
//
Sign s1 = CGAL_NTS sign((a2*a3+b2*b3)/c2-(a1*a3+b1*b3)/c1);
// We only need to make sure that w is in the correct direction
// (going from down to up)
// the original segment endpoints p1=(x1,y1) and p2=(x2,y2)
// are transformed to (v1,w1) and (v2,w2), so we need that w2 > w1
// (otherwise the result should be multiplied by -1)
const Point_2& p1 = cv.left();
const Point_2& p2 = cv.right();
FT x1 = p1.x(), y1 = p1.y(), x2 = p2.x(), y2 = p2.y();
Sign s2 = CGAL_NTS sign(-b3*x1+a3*y1-(-b3*x2+a3*y2));
return s1 * s2;
}
};
/*! Get a Compare_z_at_xy_above_3 functor object. */
Compare_z_at_xy_above_3
compare_z_at_xy_above_3_object() const
{
return Compare_z_at_xy_above_3(this);
}
/*!\brief
* Check if the surface s1 is closer/equally distanced/farther
* from the envelope with
* respect to s2 immediately below the curve c.
*/
class Compare_z_at_xy_below_3
{
protected:
const Self *parent;
public:
Compare_z_at_xy_below_3(const Self* p)
: parent(p)
{}
Comparison_result
operator()(const X_monotone_curve_2& cv,
const Xy_monotone_surface_3& surf1,
const Xy_monotone_surface_3& surf2) const
{
Comparison_result left_res =
parent->compare_z_at_xy_above_3_object()(cv, surf1, surf2);
return CGAL::opposite(left_res);
/*if (left_res == LARGER)
return SMALLER;
else if (left_res == SMALLER)
return LARGER;
else
return EQUAL;*/
}
};
/*! Get a Compare_z_at_xy_below_3 functor object. */
Compare_z_at_xy_below_3
compare_z_at_xy_below_3_object() const
{
return Compare_z_at_xy_below_3(this);
}
/***************************************************************************/
// // checks if xy-monotone surface is vertical
// class Is_vertical_3
// {
// public:
//
// bool operator()(const Xy_monotone_surface_3& s) const
// {
// return false;
// }
// };
//
// /*! Get a Is_vertical_3 functor object. */
// Is_vertical_3 is_vertical_3_object() const
// {
// return Is_vertical_3();
// }
/***************************************************************************/
// public method needed for testing
// checks if point is in the xy-range of surf
class Is_defined_over
{
public:
// checks if point is in the xy-range of surf
bool operator()(const Point_2& point,
const Xy_monotone_surface_3& surf) const
{
Kernel k;
Self parent;
// project the surface on the plane
Triangle_2 boundary = parent.project(surf);
// if surface is not vertical (i.e. boundary is not degenerate)
// check if the projected point is inside the projected boundary
if (!k.is_degenerate_2_object()(boundary))
return (!k.has_on_unbounded_side_2_object()(boundary, point));
// if surface is vertical, we check if the point is collinear
// with the projected vertices, and on one of the projected segments
// of the boundary
Point_2 v1 = k.construct_vertex_2_object()(boundary, 0);
Point_2 v2 = k.construct_vertex_2_object()(boundary, 1);
Point_2 v3 = k.construct_vertex_2_object()(boundary, 2);
if (!k.collinear_2_object()(v1, v2, point))
return false;
// enough to check 2 edges, because the 3rd is part of their union
return (k.collinear_are_ordered_along_line_2_object()(v1, point, v2) ||
k.collinear_are_ordered_along_line_2_object()(v2, point, v3));
}
};
/*! Get a Is_defined_over functor object. */
Is_defined_over is_defined_over_object() const
{
return Is_defined_over();
}
Segment_2 project (const Segment_3& seg) const
{
typedef typename Kernel::Construct_vertex_3 Construct_vertex_3;
Kernel k;
Construct_vertex_3 vertex_on = k.construct_vertex_3_object();
const Point_3 q0 = (vertex_on (seg, 0));
const Point_3 q1 = (vertex_on (seg, 1));
const Point_2 p0 (q0.x(), q0.y());
const Point_2 p1 (q1.x(), q1.y());
return (k.construct_segment_2_object() (p0, p1));
}
Point_2 project(const Point_3& obj) const
{
return Point_2(obj.x(), obj.y());
}
Triangle_2 project(const Xy_monotone_surface_3& triangle_3) const
{
const Point_3& end1 = triangle_3.vertex(0),
end2 = triangle_3.vertex(1),
end3 = triangle_3.vertex(2);
Point_2 projected_end1(end1.x(), end1.y()),
projected_end2(end2.x(), end2.y()),
projected_end3(end3.x(), end3.y());
return Triangle_2(projected_end1, projected_end2, projected_end3);
}
// triangles overlap if they lie on the same plane and intersect on it.
// this test is only needed for non-vertical triangles
bool do_overlap(const Xy_monotone_surface_3& s1,
const Xy_monotone_surface_3& s2) const
{
CGAL_precondition(s1.is_xy_monotone() && !s1.is_vertical());
CGAL_precondition(s2.is_xy_monotone() && !s2.is_vertical());
Kernel k;
if (!k.do_intersect_3_object()(static_cast<Triangle_3>(s1),
static_cast<Triangle_3>(s2)))
return false;
// check if they are coplanar
Point_3 a1 = s1.vertex(0),
b1 = s1.vertex(1),
c1 = s1.vertex(2);
Point_3 a2 = s2.vertex(0),
b2 = s2.vertex(1),
c2 = s2.vertex(2);
bool b = k.coplanar_3_object()(a1, b1, c1, a2);
if (!b) return false;
b = k.coplanar_3_object()(a1, b1, c1, b2);
if (!b) return false;
b = k.coplanar_3_object()(a1, b1, c1, c2);
return b;
}
// check whethe two xy-monotone surfaces (3D-triangles or segments)
// intersect
bool do_intersect(const Xy_monotone_surface_3& s1,
const Xy_monotone_surface_3& s2) const
{
CGAL_precondition(s1.is_xy_monotone());
CGAL_precondition(s2.is_xy_monotone());
Kernel k;
if (!s1.is_segment() && !s2.is_segment())
return k.do_intersect_3_object()(static_cast<Triangle_3>(s1),
static_cast<Triangle_3>(s2));
else if (!s1.is_segment())
return k.do_intersect_3_object()(static_cast<Triangle_3>(s1),
static_cast<Segment_3>(s2));
else if (!s2.is_segment())
return k.do_intersect_3_object()(static_cast<Segment_3>(s1),
static_cast<Triangle_3>(s2));
else
// in case of two segments, we don't use easy do-intersect test
return true;
}
// intersect two xy-monotone surfaces (3D-triangles or segments)
// if the triangles overlap, the result is empty
// the result can be a segment or a point
Object intersection(const Xy_monotone_surface_3& s1,
const Xy_monotone_surface_3& s2) const
{
CGAL_precondition(s1.is_xy_monotone());
CGAL_precondition(s2.is_xy_monotone());
Kernel k;
// first, try to intersect the bounding boxes of the triangles,
// efficiently return empty object when the triangles are faraway
if (!CGAL::do_overlap(s1.bbox(), s2.bbox()))
return Object();
// if intersecting two segment - alculate the intersection
// as in the case of dimention 2
if (s1.is_segment() && s2.is_segment())
{
Object res = intersection_of_segments(s1, s2);
return res;
}
// if both triangles lie on the same (non-vertical) plane, they overlap
// we don't care about overlaps, because they are not passed to the
// algorithm anyway, so we save the costly computation
Plane_3 p1 = s1.plane();
Plane_3 p2 = s2.plane();
if (p1 == p2 || p1 == p2.opposite())
return Object();
// calculate intersection between a triangle and the other triangle's
// supporting plane
// if there is no intersection - then the triangles have no intersection
// between them.
Object inter_obj = intersection(p1, s2);
if (inter_obj.is_empty())
return Object();
// otherwise, if the intersection in a point, we should check if it lies
// inside the first triangle
Assign_3 assign_obj = k.assign_3_object();
Point_3 inter_point;
if (assign_obj(inter_point, inter_obj))
{
Object res = intersection_on_plane_3(p1, s1, inter_point);
return res;
}
else
{
// if the intersection is a segment, we check the intersection of the
// other plane-triangle pair
Segment_3 inter_seg;
CGAL_assertion(assign_obj(inter_seg, inter_obj));
assign_obj(inter_seg, inter_obj);
inter_obj = intersection(p2, s1);
// if there is no intersection - then the triangles have no intersection
// between them.
if (inter_obj.is_empty())
return Object();
if (assign_obj(inter_point, inter_obj))
{
// if the intersection is a point, which lies on the segment,
// than it is the result,
// otherwise, empty result
if (k.has_on_3_object()(inter_seg, inter_point))
return make_object(inter_point);
else
return Object();
}
else
{
// both plane-triangle intersections are segments, which are collinear,
// and lie on the line which is the intersection of the two supporting
// planes
Segment_3 inter_seg2;
CGAL_assertion(assign_obj(inter_seg2, inter_obj));
assign_obj(inter_seg2, inter_obj);
Point_3 min1 = k.construct_min_vertex_3_object()(inter_seg),
max1 = k.construct_max_vertex_3_object()(inter_seg);
Point_3 min2 = k.construct_min_vertex_3_object()(inter_seg2),
max2 = k.construct_max_vertex_3_object()(inter_seg2);
CGAL_assertion((k.collinear_3_object()(min1, min2, max1) &&
k.collinear_3_object()(min1, max2, max1)));
// we need to find the overlapping part, if exists
Point_3 min, max;
if (k.less_xyz_3_object()(min1, min2))
min = min2;
else
min = min1;
if (k.less_xyz_3_object()(max1, max2))
max = max1;
else
max = max2;
Object res;
Comparison_result comp_res = k.compare_xyz_3_object()(min, max);
if (comp_res == EQUAL)
res = make_object(min);
else if (comp_res == SMALLER)
res = make_object(Segment_3(min, max));
// else - empty result
return res;
}
}
}
// calculate intersection between triangle & point on the same plane plane
Object intersection_on_plane_3(const Plane_3& plane,
const Xy_monotone_surface_3& triangle,
const Point_3& point) const
{
Kernel k;
CGAL_precondition( triangle.is_xy_monotone() );
CGAL_precondition( !k.is_degenerate_3_object()(plane) );
CGAL_precondition( triangle.plane() == plane ||
triangle.plane() == plane.opposite());
CGAL_precondition( k.has_on_3_object()(plane, point) );
CGAL_USE(plane);
// if the point is inside the triangle, then the point is the intersection
// otherwise there is no intersection
bool has_on;
if (triangle.is_segment())
has_on = k.has_on_3_object()(static_cast<Segment_3>(triangle), point);
else
has_on = k.has_on_3_object()(static_cast<Triangle_3>(triangle), point);
if (has_on)
return make_object(point);
else
return Object();
}
// calculate intersection between 2 segments on the same vertical plane plane
Object intersection_of_segments(const Xy_monotone_surface_3& s1,
const Xy_monotone_surface_3& s2) const
{
Kernel k;
CGAL_precondition( s1.is_xy_monotone() && s1.is_segment());
CGAL_precondition( s2.is_xy_monotone() && s2.is_segment());
// if the segments are not coplanar, they cannot intersect
if (!k.coplanar_3_object()(s1.vertex(0), s1.vertex(1),
s2.vertex(0), s2.vertex(1)))
return Object();
const Plane_3& plane = s1.plane();
if (s2.plane() != plane &&
s2.plane() != plane.opposite())
// todo: this case is not needed in the algorithm,
// so we don't implement it
return Object();
CGAL_precondition( !k.is_degenerate_3_object()(plane) );
CGAL_precondition( s2.plane() == plane ||
s2.plane() == plane.opposite());
// for simplicity, we transform the segments to the xy-plane,
// compute the intersection there, and transform it back to the 3d plane.
Point_2 v1 = plane.to_2d(s1.vertex(0)),
v2 = plane.to_2d(s1.vertex(1));
Segment_2 seg1_t(v1, v2);
Point_2 u1 = plane.to_2d(s2.vertex(0)),
u2 = plane.to_2d(s2.vertex(1));
Segment_2 seg2_t(u1, u2);
Object inter_obj = k.intersect_2_object()(seg1_t, seg2_t);
Assign_2 assign_2 = k.assign_2_object();
if (inter_obj.is_empty())
return inter_obj;
Point_2 inter_point;
Segment_2 inter_segment;
if (assign_2(inter_point, inter_obj))
return make_object(plane.to_3d(inter_point));
else
{
CGAL_assertion_code(bool b = )
assign_2(inter_segment, inter_obj);
CGAL_assertion(b);
return make_object
(Segment_3
(plane.to_3d(k.construct_vertex_2_object()(inter_segment, 0)),
plane.to_3d(k.construct_vertex_2_object()(inter_segment, 1))));
}
}
// calculate the intersection between a triangle/segment
// and a (non degenerate) plane in 3d
// the result object can be empty, a point, a segment or the original
// triangle
Object intersection(const Plane_3& pl,
const Xy_monotone_surface_3& tri) const
{
Kernel k;
CGAL_precondition( tri.is_xy_monotone() );
CGAL_precondition( !k.is_degenerate_3_object()(pl) );
if (tri.is_segment())
return k.intersect_3_object()(pl, static_cast<Segment_3>(tri));
// first, check for all 3 vertices of tri on which side of pl they lie on
int points_on_plane[3]; // contains the indices of vertices that lie
// on pl
int points_on_positive[3]; // contains the indices of vertices that lie on
// the positive side of pl
int points_on_negative[3]; // contains the indices of vertices that lie on
// the negative side of pl
int n_points_on_plane = 0;
int n_points_on_positive = 0;
int n_points_on_negative = 0;
Oriented_side side;
for (int i=0; i<3; ++i)
{
side = pl.oriented_side(tri.vertex(i));
if (side == ON_NEGATIVE_SIDE)
points_on_negative[n_points_on_negative++] = i;
else if (side == ON_POSITIVE_SIDE)
points_on_positive[n_points_on_positive++] = i;
else
points_on_plane[n_points_on_plane++] = i;
}
CGAL_assertion(n_points_on_plane +
n_points_on_positive + n_points_on_negative == 3);
// if all vertices of tri lie on the same size (positive/negative) of pl,
// there is no intersection
if (n_points_on_positive == 3 || n_points_on_negative == 3)
return Object();
// if all vertices of tri lie on pl then we return tri
if (n_points_on_plane == 3)
return make_object(tri);
// if 2 vertices lie on pl, then return the segment between them
if (n_points_on_plane == 2)
{
int point_idx1 = points_on_plane[0], point_idx2 = points_on_plane[1];
return make_object (Segment_3(tri.vertex(point_idx1),
tri.vertex(point_idx2)));
}
// if only 1 lie on pl, should check the segment opposite to it on tri
if (n_points_on_plane == 1)
{
int point_on_plane_idx = points_on_plane[0];
// if the other 2 vertices are on the same side of pl,
// then the answer is just this vertex
if (n_points_on_negative == 2 || n_points_on_positive == 2)
return make_object(tri.vertex(point_on_plane_idx));
// now it is known that one vertex is on pl, and the segment of tri
// opposite to it should intersect pl
// the segment of tri opposite of tri[point_on_plane_idx]
Segment_3 tri_segment(tri.vertex(point_on_plane_idx+1),
tri.vertex(point_on_plane_idx+2));
Object inter_result = k.intersect_3_object()(pl, tri_segment);
Point_3 inter_point;
CGAL_assertion( k.assign_3_object()(inter_point, inter_result) );
k.assign_3_object()(inter_point, inter_result);
// create the resulting segment
// (between tri[point_on_plane_idx] and inter_point)
return make_object(Segment_3(tri.vertex(point_on_plane_idx),
inter_point));
}
CGAL_assertion( n_points_on_plane == 0 );
CGAL_assertion( n_points_on_positive + n_points_on_negative == 3 );
CGAL_assertion( n_points_on_positive != 0 );
CGAL_assertion( n_points_on_negative != 0 );
// now it known that there is an intersection between 2 segments of tri
// and pl, it is also known which segments are those.
Point_3 inter_points[2];
int pos_it, neg_it, n_inter_points = 0;
for(pos_it = 0; pos_it < n_points_on_positive; ++pos_it)
for(neg_it = 0; neg_it < n_points_on_negative; ++neg_it)
{
Segment_3 seg(tri.vertex(points_on_positive[pos_it]),
tri.vertex(points_on_negative[neg_it]));
Object inter_result = k.intersect_3_object()(pl, seg);
Point_3 inter_point;
// the result of the intersection must be a point
CGAL_assertion( k.assign_3_object()(inter_point, inter_result) );
k.assign_3_object()(inter_point, inter_result);
inter_points[n_inter_points++] = inter_point;
}
CGAL_assertion( n_inter_points == 2 );
return make_object(Segment_3(inter_points[0], inter_points[1]));
}
// compare the value of s1 in p1 to the value of s2 in p2
Comparison_result
compare_z(const Point_2& p1,
const Xy_monotone_surface_3& s1,
const Point_2& p2,
const Xy_monotone_surface_3& s2)
{
CGAL_precondition(is_defined_over_object()(p1, s1));
CGAL_precondition(is_defined_over_object()(p2, s2));
Point_3 v1 = envelope_point_of_surface(p1, s1);
Point_3 v2 = envelope_point_of_surface(p2, s2);
Kernel k;
return k.compare_z_3_object()(v1, v2);
}
// find the envelope point of the surface over the given point
// precondition: the surface is defined in point
Point_3
envelope_point_of_surface(const Point_2& p,
const Xy_monotone_surface_3& s) const
{
CGAL_precondition(s.is_xy_monotone());
CGAL_precondition(is_defined_over_object()(p, s));
Point_3 point(p.x(), p.y(), 0);
// Compute the intersetion between the vertical line and the given surfaces
if (s.is_segment())
return envelope_point_of_segment(point, s);
else
{
// s is a non-vertical triangle
CGAL_assertion(!s.is_vertical());
// Construct a vertical line passing through point
Kernel k;
Direction_3 dir (0, 0, 1);
Line_3 vl = k.construct_line_3_object() (point, dir);
const Plane_3& plane = s.plane();
Object res = k.intersect_3_object()(plane, vl);
CGAL_assertion(!res.is_empty());
Point_3 ip;
CGAL_assertion(k.assign_3_object()(ip, res));
k.assign_3_object()(ip, res);
return ip;
}
}
// find the envelope point of the surface over the given point
// precondition: the surface is defined in point and is a segment
Point_3 envelope_point_of_segment(const Point_3& point,
const Xy_monotone_surface_3& s) const
{
Kernel k;
CGAL_precondition(s.is_segment());
CGAL_precondition(is_defined_over_object()(project(point), s));
// this is the vertical plane through the segment
const Plane_3& plane = s.plane();
// Construct a vertical line passing through point
Direction_3 dir (0, 0, 1);
Line_3 vl = k.construct_line_3_object() (point, dir);
// we need 2 points on this line, to be transformed to 2d,
// and preserve the direction of the envelope
Point_3 vl_point1 = k.construct_point_on_3_object()(vl, 0),
vl_point2 = k.construct_point_on_3_object()(vl, 1);
// the surface and the line are on the same plane(plane),
// so we transform them to the xy-plane, compute the intersecting point
// and transform it back to plane.
const Point_3& v1 = s.vertex(0);
const Point_3& v2 = s.vertex(1);
Point_2 t1 = plane.to_2d(v1);
Point_2 t2 = plane.to_2d(v2);
Point_2 tvl_point1 = plane.to_2d(vl_point1);
Point_2 tvl_point2 = plane.to_2d(vl_point2);
Line_2 l(tvl_point1, tvl_point2);
Segment_2 seg(t1, t2);
Object inter_obj = k.intersect_2_object()(seg, l);
Point_2 inter_point;
CGAL_assertion_code(bool is_inter_point =)
k.assign_2_object()(inter_point, inter_obj);
CGAL_assertion(is_inter_point);
return plane.to_3d(inter_point);
}
Point_2 construct_middle_point(const Point_2& p1, const Point_2& p2) const
{
Kernel k;
return k.construct_midpoint_2_object()(p1, p2);
}
Point_2 construct_middle_point(const X_monotone_curve_2& cv) const
{
Kernel k;
return k.construct_midpoint_2_object()(cv.source(), cv.target());
}
/***************************************************************************/
// for vertical decomposition
/***************************************************************************/
class Construct_vertical_2
{
public:
X_monotone_curve_2 operator()(const Point_2& p1, const Point_2& p2) const
{
return X_monotone_curve_2(p1, p2);
}
};
/*! Get a Construct_vertical_2 functor object. */
Construct_vertical_2 construct_vertical_2_object() const
{
return Construct_vertical_2();
}
Point_2 vertical_ray_shoot_2 (const Point_2& pt,
const X_monotone_curve_2& cv) const
{
CGAL_precondition(!cv.is_vertical());
typename Kernel::Segment_2 seg = cv;
Kernel k;
// If the curve contains pt, return it.
if (k.has_on_2_object() (seg, pt))
return (pt);
// Construct a vertical line passing through pt.
typename Kernel::Direction_2 dir (0, 1);
typename Kernel::Line_2 vl = k.construct_line_2_object() (pt, dir);
// Compute the intersetion between the vertical line and the given curve.
Object res = k.intersect_2_object()(seg, vl);
Point_2 ip;
bool ray_shoot_successful = k.assign_2_object()(ip, res);
if (! ray_shoot_successful)
CGAL_assertion (ray_shoot_successful);
return (ip);
}
};
/*!
* \class A representation of a triangle, as used by the
* Env_triangle_traits_3 traits-class.
*/
template <class Kernel_>
class Env_triangle_3 :
public Env_triangle_traits_3<Kernel_>::_Triangle_cached_3
{
typedef Kernel_ Kernel;
typedef typename Kernel::Triangle_3 Triangle_3;
typedef typename Kernel::Point_3 Point_3;
typedef typename Kernel::Plane_3 Plane_3;
typedef typename Kernel::Segment_3 Segment_3;
typedef typename Env_triangle_traits_3<Kernel>::_Triangle_cached_3
Base;
public:
/*!
* Default constructor.
*/
Env_triangle_3() :
Base()
{}
/*!
* Constructor from a "kernel" triangle.
* \param seg The segment.
*/
Env_triangle_3(const Triangle_3& tri) :
Base(tri)
{}
/*!
* Construct a triangle from 3 end-points.
* \param p1 The first point.
* \param p2 The second point.
* \param p3 The third point.
*/
Env_triangle_3(const Point_3 &p1, const Point_3 &p2, const Point_3 &p3) :
Base(p1, p2, p3)
{}
/*!
* Construct a triangle from a plane and 3 end-points.
* \param pl The supporting plane.
* \param p1 The first point.
* \param p2 The second point.
* \param p3 The third point.
* \pre All points must be on the supporting plane.
*/
Env_triangle_3(const Plane_3& pl,
const Point_3 &p1,
const Point_3 &p2,
const Point_3 &p3) :
Base(pl, p1, p2, p3)
{}
/*!
* Construct a segment from 2 end-points.
* \param p1 The first point.
* \param p2 The second point.
*/
Env_triangle_3(const Point_3 &p1, const Point_3 &p2) :
Base(p1, p2)
{}
/*!
* Cast to a triangle.
*/
operator Triangle_3() const
{
return (Triangle_3(this->vertex(0), this->vertex(1), this->vertex(2)));
}
/*!
* Cast to a segment (only when possible).
*/
operator Segment_3() const
{
CGAL_precondition(this->is_segment());
return (Segment_3(this->vertex(0), this->vertex(1)));
}
/*!
* Create a bounding box for the triangle.
*/
Bbox_3 bbox() const
{
Triangle_3 tri(this->vertex(0), this->vertex(1), this->vertex(2));
return (tri.bbox());
}
};
template <class Kernel>
bool
operator<(const Env_triangle_3<Kernel> &a,
const Env_triangle_3<Kernel> &b)
{
if (a.vertex(0) < b.vertex(0))
return true;
if (a.vertex(0) > b.vertex(0))
return false;
if (a.vertex(1) < b.vertex(1))
return true;
if (a.vertex(1) > b.vertex(1))
return false;
if (a.vertex(2) < b.vertex(2))
return true;
if (a.vertex(2) > b.vertex(2))
return false;
return false;
}
template <class Kernel>
bool
operator==(const Env_triangle_3<Kernel> &a,
const Env_triangle_3<Kernel> &b)
{
return (a.vertex(0) == b.vertex(0) &&
a.vertex(1) == b.vertex(1) &&
a.vertex(2) == b.vertex(2));
}
/*!
* Exporter for the triangle class used by the traits-class.
*/
template <class Kernel, class OutputStream>
OutputStream& operator<< (OutputStream& os, const Env_triangle_3<Kernel>& tri)
{
os << static_cast<typename Kernel::Triangle_3>(tri);
if (tri.is_segment())
os << " (segment)";
return (os);
}
/*!
* Importer for the triangle class used by the traits-class.
*/
template <class Kernel, class InputStream>
InputStream& operator>> (InputStream& is, Env_triangle_3<Kernel>& tri)
{
typename Kernel::Triangle_3 kernel_tri;
is >> kernel_tri;
tri = kernel_tri;
return (is);
}
} //namespace CGAL
#endif