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cgal/Envelope_3/include/CGAL/Env_triangle_traits_3.h

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// Copyright (c) 2005 Tel-Aviv University (Israel).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org).
//
// $URL$
// $Id$
// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
//
// Author(s) : Michal Meyerovitch <gorgymic@post.tau.ac.il>
// Baruch Zukerman <baruchzu@post.tau.ac.il>
// Efi Fogel <efifogel@gmail.com>
/*! \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 <vector>
#include <CGAL/enum.h>
#include <CGAL/Bbox_3.h>
#include <CGAL/Arr_segment_traits_2.h>
#include <CGAL/Envelope_3/Envelope_base.h>
namespace CGAL {
template <typename Kernel_> class Env_triangle_3;
// this traits class supports both triagles and segments in 3d
template <typename Kernel_,
typename ArrSegmentTraits = Arr_segment_traits_2<Kernel_>>
class Env_triangle_traits_3 : public ArrSegmentTraits {
public:
using Traits_2 = Arr_segment_traits_2<Kernel_>;
using Point_2 = typename Traits_2::Point_2;
using X_monotone_curve_2 = typename Traits_2::X_monotone_curve_2;
using Multiplicity = typename Traits_2::Multiplicity;
using Kernel = Kernel_;
using Self = Env_triangle_traits_3<Kernel>;
using Point_3 = typename Kernel::Point_3;
/*! \class Representation of a 3d triangle with cached data.
*/
class _Triangle_cached_3 {
public:
using Plane_3 = typename Kernel::Plane_3;
using Triangle_3 = typename Kernel::Triangle_3;
using Point_3 = typename Kernel::Point_3;
using Segment_3 = typename Kernel::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));
auto 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));
auto 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 whether the triangle is vertical.
*/
bool is_vertical() const { return (is_vert); }
/*! Check whether the surface is a segment.
*/
bool is_segment() const { return (is_seg); }
/*! Check whether 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
using Xy_monotone_surface_3 = Env_triangle_3<Kernel>;
//! type of surfaces
using Surface_3 = Xy_monotone_surface_3;
// we have a collision between the Kernel's Intersect_2 and the one
// from the segment traits
using Intersect_2 = typename Traits_2::Intersect_2;
protected:
using FT = typename Kernel::FT;
using Triangle_2 = typename Kernel::Triangle_2;
using Segment_2 = typename Kernel::Segment_2;
using Segment_3 = typename Kernel::Segment_3;
using Triangle_3 = typename Kernel::Triangle_3;
using Plane_3 = typename Kernel::Plane_3;
using Assign_2 = typename Kernel::Assign_2;
using Construct_vertex_2 = typename Kernel::Construct_vertex_2;
using Assign_3 = typename Kernel::Assign_3;
using Intersect_3 = typename Kernel::Intersect_3;
using Construct_vertex_3 = typename Kernel::Construct_vertex_3;
using Line_2 = typename Kernel::Line_2;
using Direction_2 = typename Kernel::Direction_2;
using Line_3 = typename Kernel::Line_3;
using Direction_3 = typename Kernel::Direction_3;
using Intersection_curve = std::pair<X_monotone_curve_2, Multiplicity>;
public:
/***************************************************************************/
// EnvelopeTraits_3 functors
/***************************************************************************/
/*! Subdivide a given surface into \f$xy\f$-monotone parts.
*/
class Make_xy_monotone_3 {
protected:
using Traits_3 = Env_triangle_traits_3<Kernel>;
//! The traits (in case it has state).
const Traits_3& m_traits;
/*! Constructor
* \param traits the traits
*/
Make_xy_monotone_3(const Traits_3& traits) : m_traits(traits) {}
friend class Env_triangle_traits_3<Kernel>;
public:
// create xy-monotone surfaces from a general surface
// return a past-the-end iterator
template <typename 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);
const Point_3& a2 = s.vertex(1);
const Point_3& a3 = s.vertex(2);
Point_2 b1 = m_traits.project(a1);
Point_2 b2 = m_traits.project(a2);
Point_2 b3 = m_traits.project(a3);
const Kernel& k = m_traits;
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());
const Kernel& k = m_traits;
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 <typename 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)
// and comparing it with 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();
FT w2 = a*p2.y() - b*p2.x();
FT 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 transformed 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;
};
/*! Obtain a Make_xy_monotone_3 functor object. */
Make_xy_monotone_3
make_xy_monotone_3_object() const { return Make_xy_monotone_3(*this); }
/*! Compute all planar \f$x\f$-monotone curves and possibly isolated planar
* points that form the projection of the boundary of the given
* \f$xy\f$-monotone surface s onto the \f$xy\f$-plane.
*/
class Construct_projected_boundary_2 {
protected:
using Traits_3 = Env_triangle_traits_3<Kernel>;
//! The traits (in case it has state).
const Traits_3& m_traits;
/*! Constructor
* \param traits the traits
*/
Construct_projected_boundary_2(const Traits_3& traits) : m_traits(traits) {}
friend class Env_triangle_traits_3<Kernel>;
public:
// 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 <typename 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);
const Point_3& a2 = s.vertex(1);
const Point_3& a3 = s.vertex(2);
Point_2 b1 = m_traits.project(a1);
Point_2 b2 = m_traits.project(a2);
Point_2 b3 = m_traits.project(a3);
const Kernel& k = m_traits;
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++ = std::make_pair(A, s1);
*o++ = std::make_pair(B, s2);
*o++ = std::make_pair(C, s3);
return o;
}
// s is a segment, and so is its projection
// s shouldn't be a z-vertical segment
const Point_3& a1 = s.vertex(0);
const Point_3& a2 = s.vertex(1);
Point_2 b1 = m_traits.project(a1);
Point_2 b2 = m_traits.project(a2);
CGAL_assertion(b1 != b2);
*o++ = std::make_pair(X_monotone_curve_2(b1, b2), ON_ORIENTED_BOUNDARY);
return o;
}
};
/*! Obtain a Construct_projected_boundary_curves_2 functor object. */
Construct_projected_boundary_2
construct_projected_boundary_2_object() const
{ return Construct_projected_boundary_2(*this); }
/*! compute the projection of the intersections of the \f$xy\f$-monotone
* surfaces onto the \f$xy\f$-plane,
*/
class Construct_projected_intersections_2 {
protected:
using Traits_3 = Env_triangle_traits_3<Kernel>;
//! The traits (in case it has state).
const Traits_3& m_traits;
/*! Constructor
* \param traits the traits
*/
Construct_projected_intersections_2(const Traits_3& traits) :
m_traits(traits)
{}
friend class Env_triangle_traits_3<Kernel>;
public:
// 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 <typename 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());
if (! m_traits.do_intersect(s1, s2)) return o;
const Kernel& k = m_traits;
std::optional<std::variant<Point_3, Segment_3>> inter_obj =
m_traits.intersection(s1, s2);
if (inter_obj == std::nullopt) return o;
if (const auto* point = std::get_if<Point_3>(&(*inter_obj))) {
*o++ = m_traits.project(*point);
return o;
}
const auto* curve = std::get_if<Segment_3>(&(*inter_obj));
CGAL_assertion(curve != nullptr);
Segment_2 proj_seg = m_traits.project(*curve);
if (! k.is_degenerate_2_object() (proj_seg)) {
Intersection_curve inter_cv (proj_seg, 1);
*o++ = inter_cv;
return o;
}
const Point_2& p = k.construct_point_on_2_object() (proj_seg, 0);
*o++ = p;
return o;
}
};
/*! Obtain a Construct_projected_intersections_2 functor object. */
Construct_projected_intersections_2
construct_projected_intersections_2_object() const
{ return Construct_projected_intersections_2(*this); }
/*! Determine the relative \f$z\f$-order of two given \f$xy\f$-monotone
* surfaces at the \f$xy\f$-coordinates of a given point or \f$x\f$-monotone
* curve.
*/
class Compare_z_at_xy_3 {
protected:
using Traits_3 = Env_triangle_traits_3<Kernel>;
//! The traits (in case it has state).
const Traits_3& m_traits;
/*! Constructor
* \param traits the traits
*/
Compare_z_at_xy_3(const Traits_3& traits) : m_traits(traits) {}
friend class Env_triangle_traits_3<Kernel>;
public:
// 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;
// Compute the intersetion between the vertical line and the given
// surfaces
const Kernel& k = m_traits;
Point_3 ip1 = m_traits.envelope_point_of_surface(p, surf1);
Point_3 ip2 = m_traits.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 =
m_traits.compare_z_at_xy_3_object()(cv.left(), surf1, surf2);
if (res == EQUAL) {
res = m_traits.compare_z_at_xy_3_object()(cv.right(), surf1, surf2);
if (res == EQUAL) {
Point_2 mid = m_traits.construct_middle_point(cv);
res = m_traits.compare_z_at_xy_3_object()(mid, surf1, surf2);
}
}
return res;
}
};
/*! Obtain 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); }
/*! Determine the relative \f$z\f$-order of the two given \f$xy\f$-monotone
* surfaces immediately above their projected intersection curve (a planar
* point \f$p\f$ is above an \f$x\f$-monotone curve \f$c\f$ if it is in the
* \f$x\f$-range of \f$c\f$, and lies to its left when the curve is traversed
* from its \f$xy\f$-lexicographically smaller endpoint to its larger
* endpoint).
*/
class Compare_z_at_xy_above_3 {
protected:
using Traits_3 = Env_triangle_traits_3<Kernel>;
//! The traits (in case it has state).
const Traits_3& m_traits;
/*! Constructor
* \param traits the traits
*/
Compare_z_at_xy_above_3(const Traits_3& traits) : m_traits(traits) {}
friend class Env_triangle_traits_3<Kernel>;
public:
// 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 choice 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(m_traits.compare_z_at_xy_3_object()
(cv, surf1, surf2) == EQUAL);
CGAL_precondition(m_traits.compare_z_at_xy_3_object()
(cv.source(), surf1, surf2) == EQUAL);
CGAL_precondition(m_traits.compare_z_at_xy_3_object()
(cv.target(), surf1, surf2) == EQUAL);
if (m_traits.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;
}
};
/*! Obtain 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); }
/*! Determine the relative \f$z\f$-order of the two given \f$xy\f$-monotone
* surfaces immediately below their projected intersection curve (a planar
* point \f$p\f$ is below an \f$x\f$-monotone curve \f$c\f$ if it is in the
* \f$x\f$-range of \f$c\f$, and lies to its left when the curve is traversed
* from its \f$xy\f$-lexicographically smaller endpoint to its larger
* endpoint).
*/
class Compare_z_at_xy_below_3 {
protected:
using Traits_3 = Env_triangle_traits_3<Kernel>;
//! The traits (in case it has state).
const Traits_3& m_traits;
/*! Constructor
* \param traits the traits
*/
Compare_z_at_xy_below_3(const Traits_3& traits) : m_traits(traits) {}
friend class Env_triangle_traits_3<Kernel>;
public:
//
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 =
m_traits.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;*/
}
};
/*! Obtain 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 {
protected:
using Traits_3 = Env_triangle_traits_3<Kernel>;
//! The traits (in case it has state).
const Traits_3& m_traits;
/*! Constructor
* \param traits the traits
*/
Is_defined_over(const Traits_3& traits) : m_traits(traits) {}
friend class Env_triangle_traits_3<Kernel>;
public:
// checks if point is in the xy-range of surf
bool operator()(const Point_2& point, const Xy_monotone_surface_3& surf)
const {
const Kernel& k = m_traits;
// project the surface on the plane
Triangle_2 boundary = m_traits.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(*this); }
//
Segment_2 project (const Segment_3& seg) const {
using Construct_vertex_3 = typename Kernel::Construct_vertex_3;
const Kernel& k = *this;
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);
const Point_3& end2 = triangle_3.vertex(1);
const Point_3& end3 = triangle_3.vertex(2);
Point_2 projected_end1(end1.x(), end1.y());
Point_2 projected_end2(end2.x(), end2.y());
Point_2 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());
const Kernel& k = *this;
auto do_x = k.do_intersect_3_object();
if (! do_x(static_cast<Triangle_3>(s1), static_cast<Triangle_3>(s2)))
return false;
// check if they are coplanar
Point_3 a1 = s1.vertex(0);
Point_3 b1 = s1.vertex(1);
Point_3 c1 = s1.vertex(2);
Point_3 a2 = s2.vertex(0);
Point_3 b2 = s2.vertex(1);
Point_3 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 whether 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());
const Kernel& k = *this;
auto do_x = k.do_intersect_3_object();
if (! s1.is_segment() && ! s2.is_segment())
return do_x(static_cast<Triangle_3>(s1), static_cast<Triangle_3>(s2));
else if (! s1.is_segment())
return do_x(static_cast<Triangle_3>(s1), static_cast<Segment_3>(s2));
else if (! s2.is_segment())
return do_x(static_cast<Segment_3>(s1), static_cast<Triangle_3>(s2));
else return true; // if two segments, we don't use easy do-intersect test
}
// 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
std::optional<std::variant<Point_3, Segment_3>>
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 std::nullopt;
// if intersecting two segment - alculate the intersection
// as in the case of dimension 2
if (s1.is_segment() && s2.is_segment())
return intersection_of_segments(s1, s2);
// 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 std::nullopt;
// 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.
auto inter_obj = intersection(p1, s2);
if (inter_obj == std::nullopt) return std::nullopt;
// otherwise, if the intersection in a point, we should check if it lies
// inside the first triangle
if (const Point_3* inter_point = std::get_if<Point_3>(&(*inter_obj))) {
std::optional<Point_3> res = intersection_on_plane_3(p1, s1, *inter_point);
if (res != std::nullopt) return res.value();
}
else {
// if the intersection is a segment, we check the intersection of the
// other plane-triangle pair
const Segment_3* inter_seg = std::get_if<Segment_3>(&(*inter_obj));
CGAL_assertion(inter_seg != nullptr);
auto inter_obj2 = intersection(p2, s1);
// if there is no intersection - then the triangles have no intersection
// between them.
if (inter_obj2 == std::nullopt) return std::nullopt;
if (const Point_3* inter_point = std::get_if<Point_3>(&(*inter_obj2))) {
// 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 *inter_point;
else return std::nullopt;
}
else {
// both plane-triangle intersections are segments, which are collinear,
// and lie on the line which is the intersection of the two supporting
// planes
const Segment_3* inter_seg2 = std::get_if<Segment_3>(&(*inter_obj));
CGAL_assertion(inter_seg2 != nullptr);
Point_3 min1 = k.construct_min_vertex_3_object()(*inter_seg);
Point_3 max1 = k.construct_max_vertex_3_object()(*inter_seg);
Point_3 min2 = k.construct_min_vertex_3_object()(*inter_seg2);
Point_3 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;
Comparison_result comp_res = k.compare_xyz_3_object()(min, max);
if (comp_res == EQUAL) return min;
else if (comp_res == SMALLER) return Segment_3(min, max);
// else - empty result
}
}
return std::nullopt;
}
// calculate intersection between triangle & point on the same plane plane
std::optional<Point_3>
intersection_on_plane_3(const Plane_3& plane,
const Xy_monotone_surface_3& triangle,
const Point_3& point) const {
const Kernel& k = *this;
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 point;
else return std::nullopt;
}
// calculate intersection between 2 segments on the same vertical plane plane
std::optional<std::variant<Point_3, Segment_3>>
intersection_of_segments(const Xy_monotone_surface_3& s1,
const Xy_monotone_surface_3& s2) const {
const Kernel& k = *this;
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 std::nullopt;
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 std::nullopt;
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));
Point_2 v2 = plane.to_2d(s1.vertex(1));
Segment_2 seg1_t(v1, v2);
Point_2 u1 = plane.to_2d(s2.vertex(0));
Point_2 u2 = plane.to_2d(s2.vertex(1));
Segment_2 seg2_t(u1, u2);
auto inter_obj = k.intersect_2_object()(seg1_t, seg2_t);
if (inter_obj == std::nullopt) return std::nullopt;
if (const Point_2* inter_point = std::get_if<Point_2>(&(*inter_obj)))
return plane.to_3d(*inter_point);
const Segment_2* inter_segment = std::get_if<Segment_2>(&(*inter_obj));
CGAL_assertion(inter_segment != nullptr);
auto ctr_vertex = k.construct_vertex_2_object();
return Segment_3(plane.to_3d(ctr_vertex(*inter_segment, 0)),
plane.to_3d(ctr_vertex(*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
std::optional<std::variant<Point_3, Segment_3>>
intersection(const Plane_3& pl, const Xy_monotone_surface_3& tri) const {
const Kernel& k = *this;
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 std::nullopt;
// if all vertices of tri lie on pl then we return tri
if (n_points_on_plane == 3) return 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 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 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));
auto inter_result = k.intersect_3_object()(pl, tri_segment);
const Point_3* inter_point = std::get_if<Point_3>(&(*inter_result));
CGAL_assertion( inter_point != nullptr );
// create the resulting segment
// (between tri[point_on_plane_idx] and inter_point)
return 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 n_inter_points = 0;
for (int pos_it = 0; pos_it < n_points_on_positive; ++pos_it)
for (int 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]));
auto inter_result = k.intersect_3_object()(pl, seg);
const Point_3* inter_point = std::get_if<Point_3>(&(*inter_result));
CGAL_assertion( inter_point != nullptr );
inter_points[n_inter_points++] = *inter_point;
}
CGAL_assertion( n_inter_points == 2 );
return 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();
auto res = k.intersect_3_object()(plane, vl);
CGAL_assertion(res != std::nullopt);
const Point_3* ip = std::get_if<Point_3>(&(*res));
CGAL_assertion(ip != nullptr);
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);
Point_3 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);
auto inter_obj = k.intersect_2_object()(seg, l);
const Point_2* inter_point = std::get_if<Point_2>(&(*inter_obj));
CGAL_assertion(inter_point != nullptr);
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;
const Kernel& k = *this;
// 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.
auto res = k.intersect_2_object()(seg, vl);
const Point_2* ip = std::get_if<Point_2>(&(*res));
CGAL_assertion(ip != nullptr);
return *ip;
}
};
/*! \class A representation of a triangle, as used by the
* Env_triangle_traits_3 traits-class.
*/
template <typename Kernel_>
class Env_triangle_3 : public Env_triangle_traits_3<Kernel_>::_Triangle_cached_3
{
using Kernel = Kernel_;
using Triangle_3 = typename Kernel::Triangle_3;
using Point_3 = typename Kernel::Point_3;
using Plane_3 = typename Kernel::Plane_3;
using Segment_3 = typename Kernel::Segment_3;
using Base = typename Env_triangle_traits_3<Kernel>::_Triangle_cached_3;
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 <typename 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 <typename 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 <typename Kernel_, typename 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 <typename Kernel_, typename 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
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