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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/Distance_3/include/CGAL/Distance_3/Segment_3_Segment_3.h $ // $Id: include/CGAL/Distance_3/Segment_3_Segment_3.h b26b07a1242 $ // SPDX-License-Identifier: LGPL-3.0-or-later OR LicenseRef-Commercial // // // Author(s) : Mael Rouxel-Labbé #ifndef CGAL_DISTANCE_3_SEGMENT_3_SEGMENT_3_H #define CGAL_DISTANCE_3_SEGMENT_3_SEGMENT_3_H #include #include #include namespace CGAL { namespace Distance_3 { namespace internal { template struct Segment_3_Segment_3_Result { typename K::FT x, y; typename K::FT squared_distance; }; // Using Lumelsky, 'On Fast Computation of Distance Between Line Segments' 1984 template Segment_3_Segment_3_Result squared_distance(const typename K::Segment_3& s1, const typename K::Segment_3& s2, const K& k) { typedef typename K::FT FT; typedef typename K::Point_3 Point_3; typedef typename K::Vector_3 Vector_3; typename K::Construct_vertex_3 vertex = k.construct_vertex_3_object(); typename K::Construct_vector_3 cv = k.construct_vector_3_object(); typename K::Compute_scalar_product_3 sp = k.compute_scalar_product_3_object(); typename K::Compute_squared_distance_3 sq_dist = k.compute_squared_distance_3_object(); Segment_3_Segment_3_Result res; const Point_3& p1 = vertex(s1, 0); const Point_3& q1 = vertex(s1, 1); const Point_3& p2 = vertex(s2, 0); const Point_3& q2 = vertex(s2, 1); const Vector_3 v1 = cv(p1, q1), v2 = cv(p2, q2); const Vector_3 p1p2 = cv(p1, p2); if(p1 == q1) { if(p2 == q2) { res.x = 0; res.y = 0; res.squared_distance = sq_dist(p1, p2); return res; } const FT d = - sp(v2, v2); const FT f = sp(v2, p1p2); CGAL_assertion(d < 0); res.x = 0; res.y = boost::algorithm::clamp(f/d, 0, 1); // (f - x*c) / d res.squared_distance = sq_dist(p1, p2 + res.y*v2); return res; } else if(p2 == q2) { const FT a = sp(v1, v1); const FT e = sp(v1, p1p2); CGAL_assertion(a > 0); res.y = 0; res.x = boost::algorithm::clamp(e/a, 0, 1); // (e + y*c) / a res.squared_distance = sq_dist(p1 + res.x*v1, p2); return res; } const FT a = sp(v1, v1); const FT b = - sp(v1, v2); const FT c = - b; const FT d = - sp(v2, v2); const FT e = sp(v1, p1p2); const FT f = sp(v2, p1p2); CGAL_assertion(a > 0 && d < 0); const FT det = a*d - b*c; if(is_zero(det)) res.x = 0; else res.x = boost::algorithm::clamp((e*d - b*f) / det, 0, 1); FT xc = res.x*c; // res.y = (f - xc) / d, by definition, but building it up more efficiently if(f > xc) // y < 0 <=> f - xc / d < 0 <=> f - xc > 0 (since d is -||v2||) { res.y = 0; res.x = boost::algorithm::clamp(e/a, 0, 1); // (e + y*c) / a } else // y >= 0 { FT n = f - xc; // delay the division by d if(n < d) // y > 1 <=> n / d > 1 <=> n < d (once again, important to note that d is negative!) { res.y = 1; res.x = boost::algorithm::clamp((e + c) / a, 0, 1); // (e + y*c) / a } else // 0 <= y <= 1 { res.y = n / d; } } CGAL_postcondition(FT(0) <= res.x && res.x <= FT(1)); CGAL_postcondition(FT(0) <= res.y && res.y <= FT(1)); res.squared_distance = sq_dist(p1 + res.x*v1, p2 + res.y*v2); CGAL_postcondition(res.squared_distance >= FT(0)); return res; } } // namespace internal } // namespace Distance_3 namespace internal { template typename K::FT squared_distance_parallel(const typename K::Segment_3& seg1, const typename K::Segment_3& seg2, const K& k) { typedef typename K::Vector_3 Vector_3; typename K::Compute_squared_distance_3 sq_dist = k.compute_squared_distance_3_object(); const Vector_3 dir1 = seg1.direction().vector(); const Vector_3 dir2 = seg2.direction().vector(); if(same_direction(dir1, dir2, k)) { if(!is_acute_angle(seg1.source(), seg1.target(), seg2.source(), k)) return sq_dist(seg1.target(), seg2.source(), k); if(!is_acute_angle(seg1.target(), seg1.source(), seg2.target(), k)) return sq_dist(seg1.source(), seg2.target(), k); } else { if(!is_acute_angle(seg1.source(), seg1.target(), seg2.target(), k)) return sq_dist(seg1.target(), seg2.target(), k); if(!is_acute_angle(seg1.target(), seg1.source(), seg2.source(), k)) return sq_dist(seg1.source(), seg2.source(), k); } return sq_dist(seg2.source(), seg1.supporting_line(), k); } template typename K::FT squared_distance(const typename K::Segment_3& seg1, const typename K::Segment_3& seg2, const K& k) { Distance_3::internal::Segment_3_Segment_3_Result res = Distance_3::internal::squared_distance(seg1, seg2, k); return res.squared_distance; } template typename K::Comparison_result compare_squared_distance(const typename K::Segment_3& s1, const typename K::Segment_3& s2, const K& k, const typename K::FT& d2) { typedef typename K::FT FT; typedef typename K::Point_3 Point_3; typedef typename K::Vector_3 Vector_3; typename K::Construct_vertex_3 vertex = k.construct_vertex_3_object(); typename K::Construct_vector_3 cv = k.construct_vector_3_object(); typename K::Compute_scalar_product_3 sp = k.compute_scalar_product_3_object(); typename K::Compare_squared_distance_3 csq_dist = k.compare_squared_distance_3_object(); /* The content of this function is very similar with the one above, the difference is we can exit earlier if the supporting line are farther than d since we do not need the exact distance. */ #if 1 const Point_3& p1 = vertex(s1, 0); const Point_3& q1 = vertex(s1, 1); const Point_3& p2 = vertex(s2, 0); const Point_3& q2 = vertex(s2, 1); const Vector_3 v1 = cv(p1, q1), v2 = cv(p2, q2); const Vector_3 p1p2 = cv(p1, p2); // If degenerate segment, compare the distance between the point and the other segment if(p1 == q1) if(p2 == q2) return csq_dist(p1,p2,d2); else return csq_dist(p1,s2,d2); else if(p2 == q2) return csq_dist(s1,p2,d2); // Compare first the distance between the lines, if larger we can exit early typename K::Comparison_result res_ll=csq_dist(s1.supporting_line(), s2.supporting_line(), d2); if(certainly(res_ll==LARGER)) return LARGER; // Compute the distance between the segments // TODO some factorization with above function? only the last case is different const FT a = sp(v1, v1); const FT b = - sp(v1, v2); const FT c = - b; const FT d = - sp(v2, v2); const FT e = sp(v1, p1p2); const FT f = sp(v2, p1p2); CGAL_assertion(a > 0 && d < 0); const FT det = a*d - b*c; FT res_x; if(is_zero(det)) res_x = 0; else res_x = boost::algorithm::clamp((e*d - b*f) / det, 0, 1); FT xc = res_x*c; // res.y = (f - xc) / d, by definition, but building it up more efficiently if(f > xc) // y < 0 <=> f - xc / d < 0 <=> f - xc > 0 (since d is -||v2||) { // res_y = 0; res_x = boost::algorithm::clamp(e/a, 0, 1); // (e + y*c) / a return csq_dist(p1+res_x*v1,p2, d2); } else // y >= 0 { FT n = f - xc; // delay the division by d if(n < d) // y > 1 <=> n / d > 1 <=> n < d (once again, important to note that d is negative!) { // res_y = 1; res_x = boost::algorithm::clamp((e + c) / a, 0, 1); // (e + y*c) / a return csq_dist(p1+res_x*v1,p2+v2, d2); } else if(res_x==0 || res_x==1) // 0 <= y <= 1 { FT res_y = n / d; return csq_dist(p1 + res_x*v1, p2 + res_y*v2, d2); } else { //Already computed by distance line line return res_ll; } } #else // Faster with Simple_cartesian, a bit slower with EPICK or EPECK specifically if d2 is small return ::CGAL::compare(squared_distance(s1, s2 ,k), d2); #endif } } // namespace internal } // namespace CGAL #endif // CGAL_DISTANCE_3_SEGMENT_3_SEGMENT_3_H