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// Copyright (c) 1998-2021
// Utrecht University (The Netherlands),
// ETH Zurich (Switzerland),
// INRIA Sophia-Antipolis (France),
// Max-Planck-Institute Saarbruecken (Germany),
// and Tel-Aviv University (Israel). All rights reserved.
//
// This file is part of CGAL (www.cgal.org)
//
// $URL: https://github.com/CGAL/cgal/blob/v6.1.1/Distance_3/include/CGAL/Distance_3/Triangle_3_Triangle_3.h $
// $Id: include/CGAL/Distance_3/Triangle_3_Triangle_3.h 08b27d3db14 $
// SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial
//
//
// Author(s) : Mael Rouxel-Labbé
#ifndef CGAL_DISTANCE_3_TRIANGLE_3_TRIANGLE_3_H
#define CGAL_DISTANCE_3_TRIANGLE_3_TRIANGLE_3_H
#include <CGAL/Distance_3/Point_3_Point_3.h>
#include <CGAL/Distance_3/Segment_3_Segment_3.h>
#include <CGAL/Triangle_3.h>
namespace CGAL {
namespace Distance_3 {
namespace internal {
template <typename K>
std::pair<Segment_3_Segment_3_Result<K>, bool>
test_edge_pair(const typename K::Point_3& p1,
const typename K::Point_3& q1,
const typename K::Point_3& r1,
const typename K::Point_3& p2,
const typename K::Point_3& q2,
const typename K::Point_3& r2,
const K& k,
typename K::FT& global_min_sqd,
bool& are_triangles_known_to_be_disjoint)
{
typedef typename K::FT FT;
typedef typename K::Point_3 Point_3;
typedef typename K::Vector_3 Vector_3;
typename K::Compute_scalar_product_3 scalar_product = k.compute_scalar_product_3_object();
typename K::Construct_segment_3 segment = k.construct_segment_3_object();
typename K::Construct_scaled_vector_3 scale_vector = k.construct_scaled_vector_3_object();
typename K::Construct_vector_3 vector = k.construct_vector_3_object();
typename K::Construct_translated_point_3 translate = k.construct_translated_point_3_object();
Distance_3::internal::Segment_3_Segment_3_Result<K> res =
internal::squared_distance(segment(p1, q1), segment(p2, q2), k);
if(res.squared_distance <= global_min_sqd)
global_min_sqd = res.squared_distance;
else
return std::make_pair(res, false);
const Vector_3 v1 = vector(p1, q1), v2 = vector(p2, q2);
const Point_3 m1 = translate(p1, scale_vector(v1, res.x));
const Point_3 m2 = translate(p2, scale_vector(v2, res.y));
const Vector_3 vr1 = vector(m1, r1), vr2 = vector(m2, r2);
const Vector_3 n = vector(m1, m2);
const FT sp_r1 = scalar_product(vr1, n);
const FT sp_r2 = scalar_product(vr2, n);
const bool is_r1_closer = (sp_r1 > 0); // Plane_3{m1, n}.has_on_positive_side(r1);
const bool is_r2_closer = (sp_r2 < 0); // Plane_3{m2, -n}.has_on_positive_side(r2);
const bool is_best_pair = !is_r1_closer && !is_r2_closer;
// Even if it is not the best pair, one may be able to deduce if the triangles do not intersect
// by checking if there is a void space between the planes orthogonal to the vector realizing
// the min distance between the edges and passing through the third points.
if(!is_best_pair)
{
FT separating_distance = res.squared_distance;
if(is_r1_closer)
separating_distance -= sp_r1;
if(is_r2_closer)
separating_distance += sp_r2;
if(separating_distance > 0)
are_triangles_known_to_be_disjoint = true;
}
return std::make_pair(res, is_best_pair);
}
template <typename K>
std::pair<typename K::FT, bool>
test_vertex_triangle(const typename K::Triangle_3& tr1,
const typename K::Triangle_3& tr2,
const K& k,
bool& are_triangles_known_to_be_disjoint)
{
typedef typename K::FT FT;
typedef typename K::Point_3 Point_3;
typedef typename K::Vector_3 Vector_3;
typename K::Compute_scalar_product_3 scalar_product = k.compute_scalar_product_3_object();
typename K::Construct_cross_product_vector_3 cross_product = k.construct_cross_product_vector_3_object();
typename K::Construct_vector_3 vector = k.construct_vector_3_object();
typename K::Construct_vertex_3 vertex = k.construct_vertex_3_object();
const Point_3& p1 = vertex(tr1, 0);
const Point_3& q1 = vertex(tr1, 1);
const Point_3& r1 = vertex(tr1, 2);
const Point_3& p2 = vertex(tr2, 0);
const Point_3& q2 = vertex(tr2, 1);
const Point_3& r2 = vertex(tr2, 2);
const Vector_3 p2q2 = vector(p2, q2);
const Vector_3 p2r2 = vector(p2, r2);
const Vector_3 n2 = cross_product(p2q2, p2r2);
if(scalar_product(n2, n2) == FT(0))
return std::make_pair(0, false);
std::array<FT, 3> sps = { scalar_product(vector(p2, p1), n2),
scalar_product(vector(p2, q1), n2),
scalar_product(vector(p2, r1), n2) };
// All the vertices of tr1 must be on the same side of tr2
// Coplanarity is tolerated, so '1' and '0' should be allowed, but not '1' and '-1'
if(CGAL::sign(sps[0]) == - CGAL::sign(sps[1]) || CGAL::sign(sps[1]) == - CGAL::sign(sps[2]))
return std::make_pair(0, false);
std::for_each(sps.begin(), sps.end(), [](FT& v) { v = abs(v); });
const auto min_pos = std::min_element(sps.begin(), sps.end());
const std::size_t min_id = static_cast<std::size_t>(std::distance(sps.begin(), min_pos));
if(sps[min_id] > 0)
are_triangles_known_to_be_disjoint = true;
const Point_3& x1 = vertex(tr1, static_cast<int>(min_id));
if(CGAL::internal::on_left_of_triangle_edge(x1, n2, p2, q2, k) &&
CGAL::internal::on_left_of_triangle_edge(x1, n2, q2, r2, k) &&
CGAL::internal::on_left_of_triangle_edge(x1, n2, r2, p2, k))
{
// the projection of `x1` is inside the triangle
return std::make_pair(CGAL::internal::squared_distance_to_plane(n2, vector(p2, x1), k), true);
}
return std::make_pair(0, false);
}
} // namespace internal
} // namespace Distance_3
namespace internal {
template <typename K>
typename K::FT
squared_distance(const typename K::Triangle_3& tr1,
const typename K::Triangle_3& tr2,
const K& k)
{
typedef typename K::FT FT;
typename K::Construct_vertex_3 vertex = k.construct_vertex_3_object();
// ideally just limits<FT>::infinity|max(), but it is not available for exact NTs...
FT global_min_sqd = squared_distance(vertex(tr1, 0), vertex(tr2, 0));
bool are_triangles_known_to_be_disjoint = false;
std::pair<Distance_3::internal::Segment_3_Segment_3_Result<K>, bool> ss_res;
for(int i=0; i<3; ++i)
{
for(int j=0; j<3; ++j)
{
ss_res = Distance_3::internal::test_edge_pair(
vertex(tr1, i%3), vertex(tr1, (i+1)%3), vertex(tr1, (i+2)%3),
vertex(tr2, j%3), vertex(tr2, (j+1)%3), vertex(tr2, (j+2)%3), k,
global_min_sqd, are_triangles_known_to_be_disjoint);
if(ss_res.second)
return ss_res.first.squared_distance;
}
}
// Failed to find a minimum between segment pairs, explore vertex-triangle distances
#if 1
std::pair<FT, bool> pt_res =
Distance_3::internal::test_vertex_triangle(tr1, tr2, k, are_triangles_known_to_be_disjoint);
if(pt_res.second)
return pt_res.first;
pt_res = Distance_3::internal::test_vertex_triangle(tr2, tr1, k, are_triangles_known_to_be_disjoint);
if(pt_res.second)
return pt_res.first;
if(are_triangles_known_to_be_disjoint)
return global_min_sqd;
else
return 0;
#else // A tiny bit less efficient, but a lot clearer!
// @todo does not handle degenerate inputs
if(!are_triangles_known_to_be_disjoint && CGAL::do_intersect(tr1, tr2))
return 0;
FT sqd_p1 = CGAL::squared_distance(vertex(tr1, 0), tr2);
FT sqd_q1 = CGAL::squared_distance(vertex(tr1, 1), tr2);
FT sqd_r1 = CGAL::squared_distance(vertex(tr1, 2), tr2);
FT sqd_p2 = CGAL::squared_distance(vertex(tr2, 0), tr1);
FT sqd_q2 = CGAL::squared_distance(vertex(tr2, 1), tr1);
FT sqd_r2 = CGAL::squared_distance(vertex(tr2, 2), tr1);
const FT m = (std::min)({sqd_p1, sqd_q1, sqd_r1, sqd_p2, sqd_q2, sqd_r2});
return m;
#endif
}
template <typename K>
typename K::Comparison_result
compare_squared_distance_disjoint(const typename K::Triangle_3& tr1,
const typename K::Triangle_3& tr2,
const K& k,
const typename K::FT& d2)
{
typedef typename K::Segment_3 Segment_3;
typename K::Construct_vertex_3 vertex = k.construct_vertex_3_object();
typename K::Compare_squared_distance_3 csq_dist = k.compare_squared_distance_3_object();
typename K::Comparison_result res(LARGER);
// The tiangle are supposed to be disjoint
assert(!do_intersect(tr1, tr2));
for(int i=0; i<3; ++i)
{
//Compare the distance between edges
for(int j=0; j<3; ++j)
{
typename K::Comparison_result temp_res_ss=csq_dist(Segment_3(vertex(tr1, i%3), vertex(tr1, (i+1)%3)),Segment_3(vertex(tr2, j%3), vertex(tr2, (j+1)%3)),d2);
if(certainly(temp_res_ss==SMALLER))
return SMALLER;
res=smaller_of(res, temp_res_ss);
}
//Compare the distance between vertices and triangles
typename K::Comparison_result temp_res_v_pl= csq_dist(vertex(tr1, i), tr2,d2);
if(certainly(temp_res_v_pl==SMALLER))
return SMALLER;
res=smaller_of(res, temp_res_v_pl);
temp_res_v_pl= csq_dist(vertex(tr2, i), tr1,d2);
if(certainly(temp_res_v_pl==SMALLER))
return SMALLER;
res=smaller_of(res, temp_res_v_pl);
}
return res;
}
template <typename K>
typename K::Comparison_result
compare_squared_distance(const typename K::Triangle_3& tr1,
const typename K::Triangle_3& tr2,
const K& k,
const typename K::FT& d2){
// sq_dist and csq_dist does not exist for Segment-Triangle, sq supports degenerate triangle
if(tr1.is_degenerate() || tr2.is_degenerate())
return ::CGAL::compare(squared_distance(tr1,tr2, k), d2);
if(do_intersect(tr1, tr2))
return ::CGAL::compare(typename K::FT(0), d2);
return compare_squared_distance_disjoint(tr1, tr2, k, d2);
}
} // namespace internal
} // namespace CGAL
#endif // CGAL_DISTANCE_3_TRIANGLE_3_TRIANGLE_3_H
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