void shortest_path_regular_tetrahedron()
{
typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;
typedef CGAL::Polyhedron_3<Kernel, CGAL::Polyhedron_items_with_id_3> Polyhedron_3;
typedef CGAL::Surface_mesh_shortest_path_traits<Kernel, Polyhedron_3> Traits;
typedef Traits::Barycentric_coordinate Barycentric_coordinate;
typedef Traits::FT FT;
typedef boost::graph_traits<Polyhedron_3> Graph_traits;
typedef Graph_traits::vertex_iterator vertex_iterator;
typedef Graph_traits::face_descriptor face_descriptor;
typedef Graph_traits::face_iterator face_iterator;
typedef CGAL::Surface_mesh_shortest_path<Traits> Surface_mesh_shortest_path;
Traits traits;
Traits::Construct_barycentric_coordinate construct_barycentric_coordinate(traits.construct_barycentric_coordinate_object());
Polyhedron_3 P;
CGAL::test::make_regular_tetrahedron(P);
CGAL::set_halfedgeds_items_id(P);
Barycentric_coordinate b = construct_barycentric_coordinate(FT(1.0) / FT(3.0), FT(1.0) / FT(3.0), FT(1.0) / FT(3.0));
face_iterator startFace;
face_iterator endFace;
boost::tie(startFace,endFace) = CGAL::faces(P);
face_descriptor firstFace = *startFace;
Surface_mesh_shortest_path shortestPaths(P, traits);
//shortestPaths.m_debugOutput = true;
shortestPaths.add_source_point(firstFace, b);
shortestPaths.build_sequence_tree();
vertex_iterator currentVertex;
vertex_iterator endVertex;
Kernel::FT sideLength = Kernel::FT(2.0);
Kernel::FT halfSideLength = sideLength / Kernel::FT(2.0);
Kernel::FT triangleHeight = CGAL::sqrt((sideLength*sideLength) - (halfSideLength*halfSideLength));
for (boost::tie(currentVertex, endVertex) = CGAL::vertices(P); currentVertex != endVertex; ++currentVertex)
{
if ((*currentVertex)->point().y()==-1)
{
CHECK_CLOSE(shortestPaths.shortest_distance_to_source_points(*currentVertex).first, Kernel::FT((triangleHeight * Kernel::FT(4.0)) / Kernel::FT(3.0)), Kernel::FT(0.000001));
}
else
{
CHECK_CLOSE(shortestPaths.shortest_distance_to_source_points(*currentVertex).first, Kernel::FT((triangleHeight * Kernel::FT(2.0)) / Kernel::FT(3.0)), Kernel::FT(0.000001));
}
}
}
void Scene_polyhedron_shortest_path_item::get_as_edge_point(Scene_polyhedron_shortest_path_item::Face_location& inOutLocation)
{
size_t minIndex = 0;
FT minCoord(inOutLocation.second[0]);
for (size_t i = 1; i < 3; ++i)
{
if (minCoord > inOutLocation.second[i])
{
minIndex = i;
minCoord = inOutLocation.second[i];
}
}
// The nearest edge is that of the two non-minimal barycentric coordinates
size_t nearestEdge[2];
size_t current = 0;
for (size_t i = 0; i < 3; ++i)
{
if (i != minIndex)
{
nearestEdge[current] = i;
++current;
}
}
Construct_barycentric_coordinate construct_barycentric_coordinate;
Point_3 trianglePoints[3] = {
m_shortestPaths->point(inOutLocation.first, construct_barycentric_coordinate(FT(1.0), FT(0.0), FT(0.0))),
m_shortestPaths->point(inOutLocation.first, construct_barycentric_coordinate(FT(0.0), FT(1.0), FT(0.0))),
m_shortestPaths->point(inOutLocation.first, construct_barycentric_coordinate(FT(0.0), FT(0.0), FT(1.0))),
};
CGAL::Surface_mesh_shortest_paths_3::Parametric_distance_along_segment_3<Surface_mesh_shortest_path_traits> parametricDistanceSegment3;
Point_3 trianglePoint = m_shortestPaths->point(inOutLocation.first, inOutLocation.second);
FT distanceAlongSegment = parametricDistanceSegment3(trianglePoints[nearestEdge[0]], trianglePoints[nearestEdge[1]], trianglePoint);
FT coords[3] = { FT(0.0), FT(0.0), FT(0.0), };
coords[nearestEdge[1]] = distanceAlongSegment;
coords[nearestEdge[0]] = FT(1.0) - distanceAlongSegment;
inOutLocation.second = construct_barycentric_coordinate(coords[0], coords[1], coords[2]);
}
void Scene_polyhedron_shortest_path_item::get_as_vertex_point(Scene_polyhedron_shortest_path_item::Face_location& inOutLocation)
{
size_t maxIndex = 0;
FT maxCoord(inOutLocation.second[0]);
for (size_t i = 1; i < 3; ++i)
{
if (inOutLocation.second[i] > maxCoord)
{
maxIndex = i;
maxCoord = inOutLocation.second[i];
}
}
FT coords[3] = { FT(0.0), FT(0.0), FT(0.0), };
coords[maxIndex] = FT(1.0);
Construct_barycentric_coordinate construct_barycentric_coordinate;
inOutLocation.second = construct_barycentric_coordinate(coords[0], coords[1], coords[2]);
}
bool Scene_polyhedron_shortest_path_item::deferred_load(Scene_polyhedron_item* polyhedronItem, Scene_interface* sceneInterface, Messages_interface* messages, QMainWindow* mainWindow)
{
initialize(polyhedronItem, sceneInterface, messages, mainWindow);
std::ifstream inFile(m_deferredLoadFilename.c_str());
if (!inFile)
{
return false;
}
m_shortestPaths->clear();
std::vector<face_descriptor> listOfFaces;
listOfFaces.reserve(CGAL::num_faces(*polyhedron()));
face_iterator current, end;
for (boost::tie(current, end) = CGAL::faces(*polyhedron()); current != end; ++current)
{
listOfFaces.push_back(*current);
}
std::string line;
std::size_t faceId;
Barycentric_coordinate location;
Construct_barycentric_coordinate construct_barycentric_coordinate;
while (std::getline(inFile, line))
{
std::istringstream lineStream(line);
FT coords[3];
lineStream >> faceId >> coords[0] >> coords[1] >> coords[2];
location = construct_barycentric_coordinate(coords[0], coords[1], coords[2]);
// std::cout << "Read in face: " << faceId << " , " << location << std::endl;
m_shortestPaths->add_source_point(listOfFaces[faceId], location);
}
return true;
}
void test_simple_saddle_vertex_mesh()
{
typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;
typedef CGAL::Polyhedron_3<Kernel, CGAL::Polyhedron_items_with_id_3> Polyhedron_3;
typedef CGAL::Surface_mesh_shortest_path_traits<Kernel, Polyhedron_3> Traits;
typedef Traits::Barycentric_coordinate Barycentric_coordinate;
typedef Traits::FT FT;
typedef Traits::Point_3 Point_3;
typedef Traits::Point_2 Point_2;
typedef Traits::Triangle_3 Triangle_3;
typedef Traits::Triangle_2 Triangle_2;
typedef Traits::Segment_2 Segment_2;
typedef boost::graph_traits<Polyhedron_3> Graph_traits;
typedef Graph_traits::vertex_descriptor vertex_descriptor;
typedef Graph_traits::vertex_iterator vertex_iterator;
typedef Graph_traits::halfedge_descriptor halfedge_descriptor;
typedef Graph_traits::face_descriptor face_descriptor;
typedef CGAL::Surface_mesh_shortest_path<Traits> Surface_mesh_shortest_path;
typedef boost::property_map<Polyhedron_3, CGAL::vertex_point_t>::type VPM;
Traits traits;
Traits::Compute_squared_distance_3 compute_squared_distance_3(traits.compute_squared_distance_3_object());
Traits::Compute_squared_distance_2 compute_squared_distance_2(traits.compute_squared_distance_2_object());
Traits::Construct_triangle_3_along_segment_2_flattening flatten_triangle_3_along_segment_2(traits.construct_triangle_3_along_segment_2_flattening_object());
Traits::Construct_barycentric_coordinate construct_barycentric_coordinate(traits.construct_barycentric_coordinate_object());
std::ifstream inFile("data/saddle_vertex_mesh.off");
Polyhedron_3 P;
inFile >> P;
inFile.close();
CGAL::set_halfedgeds_items_id(P);
vertex_iterator startVertex;
vertex_iterator endVertex;
boost::tie(startVertex, endVertex) = CGAL::vertices(P);
vertex_iterator currentVertex = startVertex;
++currentVertex;
vertex_descriptor rootSearchVertex = *currentVertex;
face_descriptor currentFace = CGAL::face(CGAL::halfedge(rootSearchVertex, P), P);
size_t vertexIndex = CGAL::test::face_vertex_index(currentFace, rootSearchVertex, P);
Barycentric_coordinate baryCoord = construct_barycentric_coordinate(vertexIndex == 0 ? FT(1.0) : FT(0.0), vertexIndex == 1 ? FT(1.0) : FT(0.0), vertexIndex == 2 ? FT(1.0) : FT(0.0));
Surface_mesh_shortest_path shortestPaths(P, traits);
//shortestPaths.m_debugOutput = true;
Surface_mesh_shortest_path::Source_point_iterator firstSourcePoint = shortestPaths.add_source_point(currentFace, baryCoord);
shortestPaths.build_sequence_tree();
VPM vpm = CGAL::get(CGAL::vertex_point, P);
Point_3 vertexLocations[8];
vertex_descriptor vertexHandles[8];
currentVertex = startVertex;
for (size_t i = 0; i < 8; ++i)
{
vertexHandles[i] = *currentVertex;
vertexLocations[i] = vpm[*currentVertex];
++currentVertex;
}
FT distanceToBottom = CGAL::sqrt(compute_squared_distance_3(vertexLocations[1], vertexLocations[0]));
FT largerSideLength = CGAL::sqrt(compute_squared_distance_3(vertexLocations[1], vertexLocations[2]));
FT distanceToSaddle = CGAL::sqrt(compute_squared_distance_3(vertexLocations[1], vertexLocations[4]));
FT shorterSideLength = CGAL::sqrt(compute_squared_distance_3(vertexLocations[4], vertexLocations[6]));
Triangle_3 lower(vertexLocations[6], vertexLocations[4], vertexLocations[1]);
Triangle_3 upper(vertexLocations[4], vertexLocations[6], vertexLocations[7]);
Segment_2 base(Point_2(CGAL::ORIGIN), Point_2(shorterSideLength, FT(0.0)));
Triangle_2 flatLower(flatten_triangle_3_along_segment_2(lower, 0, base));
Triangle_2 flatUpper(flatten_triangle_3_along_segment_2(upper, 0, Segment_2(base[1], base[0])));
FT distanceToApex = CGAL::sqrt(compute_squared_distance_2(flatLower[2], flatUpper[2]));
FT expectedDistances[8] =
{
distanceToBottom, // a vertex of the larger tetrahedron
FT(0.0), // the initial vertex
largerSideLength, // a vertex of the larger tetrahedron
largerSideLength, // a vertex of the larger tetrahedron
distanceToSaddle, // direct line of sight from root
distanceToSaddle + shorterSideLength, // around the corner from a pseudo-source (not in direct line of geodesic sight)
distanceToSaddle, // direct line of sight from root
distanceToApex,
};
currentVertex = startVertex;
for (size_t i = 0; i < 8; ++i)
{
CHECK_CLOSE(shortestPaths.shortest_distance_to_source_points(*currentVertex).first, expectedDistances[i], Kernel::FT(0.0001));
++currentVertex;
}
// test the edge sequence reporting
CGAL::test::Edge_sequence_collector<Traits> collector(P);
shortestPaths.shortest_path_sequence_to_source_points(vertexHandles[5], collector);
CHECK_EQUAL(collector.m_sequence.size(), 3u);
CHECK_EQUAL(collector.m_sequence[1].type, CGAL::test::SEQUENCE_ITEM_VERTEX);
assert(collector.m_sequence[1].index == 4 || collector.m_sequence[1].index == 6);
CHECK_EQUAL(collector.m_sequence[2].type, CGAL::test::SEQUENCE_ITEM_VERTEX);
CHECK_EQUAL(collector.m_sequence[2].index, 1u);
collector.m_sequence.clear();
typedef boost::property_map<Polyhedron_3, CGAL::halfedge_external_index_t>::type HalfedgeIndexMap;
HalfedgeIndexMap halfedgeIndexMap(CGAL::get(CGAL::halfedge_external_index, P));
shortestPaths.shortest_path_sequence_to_source_points(vertexHandles[7], collector);
CHECK_EQUAL(collector.m_sequence.size(), 3u);
CHECK_EQUAL(collector.m_sequence[1].type, CGAL::test::SEQUENCE_ITEM_EDGE);
CHECK_EQUAL(collector.m_sequence[1].index, halfedgeIndexMap[CGAL::halfedge(vertexHandles[4], vertexHandles[6], P).first]);
CHECK_CLOSE(collector.m_sequence[1].edgeAlpha, FT(0.5), FT(0.0001));
CHECK_EQUAL(collector.m_sequence[2].type, CGAL::test::SEQUENCE_ITEM_VERTEX);
CHECK_EQUAL(collector.m_sequence[2].index, 1u);
// Now test an internal face location sequence
halfedge_descriptor firstCrossing = CGAL::halfedge(vertexHandles[4], vertexHandles[7], P).first;
size_t edgeIndex = CGAL::internal::edge_index(firstCrossing, P);
Barycentric_coordinate location = construct_barycentric_coordinate(0.25, 0.5, 0.25);
collector.m_sequence.clear();
shortestPaths.shortest_path_sequence_to_source_points(CGAL::face(firstCrossing, P), construct_barycentric_coordinate(location[edgeIndex], location[(edgeIndex + 1) % 3], location[(edgeIndex + 2) % 3]), collector);
CHECK_EQUAL(collector.m_sequence.size(), 4u);
CHECK_EQUAL(collector.m_sequence[1].type, CGAL::test::SEQUENCE_ITEM_EDGE);
CHECK_EQUAL(collector.m_sequence[1].index, halfedgeIndexMap[firstCrossing]);
CHECK_EQUAL(collector.m_sequence[2].type, CGAL::test::SEQUENCE_ITEM_EDGE);
CHECK_EQUAL(collector.m_sequence[2].index, halfedgeIndexMap[CGAL::halfedge(vertexHandles[4], vertexHandles[6], P).first]);
CHECK_EQUAL(collector.m_sequence[3].type, CGAL::test::SEQUENCE_ITEM_VERTEX);
CHECK_EQUAL(collector.m_sequence[3].index, 1u);
// Now test with 2 source vertices
currentVertex = startVertex;
for (size_t i = 0; i < 5; ++i)
{
++currentVertex;
}
vertex_descriptor rootSearchVertex2 = *currentVertex;
face_descriptor currentFace2 = CGAL::face(CGAL::halfedge(rootSearchVertex2, P), P);
size_t vertexIndex2 = CGAL::test::face_vertex_index(currentFace2, rootSearchVertex2, P);
Barycentric_coordinate baryCoord2 = construct_barycentric_coordinate(vertexIndex2 == 0 ? FT(1.0) : FT(0.0), vertexIndex2 == 1 ? FT(1.0) : FT(0.0), vertexIndex2 == 2 ? FT(1.0) : FT(0.0));
Surface_mesh_shortest_path::Source_point_iterator secondSourcePoint = shortestPaths.add_source_point(currentFace2, baryCoord2);
shortestPaths.build_sequence_tree();
FT distanceToApexFrom2 = CGAL::sqrt(compute_squared_distance_3(vertexLocations[5], vertexLocations[7]));
Triangle_3 lower2(vertexLocations[2], vertexLocations[3], vertexLocations[5]);
Triangle_3 upper2(vertexLocations[3], vertexLocations[2], vertexLocations[0]);
Segment_2 base2(Point_2(CGAL::ORIGIN), Point_2(largerSideLength, FT(0.0)));
Triangle_2 flatLower2(flatten_triangle_3_along_segment_2(lower2, 0, base2));
Triangle_2 flatUpper2(flatten_triangle_3_along_segment_2(upper2, 0, Segment_2(base2[1], base2[0])));
FT distanceToBottom2 = CGAL::sqrt(compute_squared_distance_2(flatLower2[2], flatUpper2[2]));
FT expectedDistances2[8] =
{
distanceToBottom2, // a vertex of the larger tetrahedron
FT(0.0), // an initial vertex
distanceToSaddle, // a vertex of the larger tetrahedron
distanceToSaddle, // a vertex of the larger tetrahedron
shorterSideLength, // direct line of sight from root
FT(0.0), // around the corner from a pseudo-source (not in direct line of geodesic sight)
shorterSideLength, // direct line of sight from root
distanceToApexFrom2,
};
Surface_mesh_shortest_path::Source_point_iterator expectedSources2[8] =
{
secondSourcePoint,
firstSourcePoint,
secondSourcePoint,
secondSourcePoint,
secondSourcePoint,
secondSourcePoint,
secondSourcePoint,
secondSourcePoint,
};
currentVertex = startVertex;
for (size_t i = 0; i < 8; ++i)
{
Surface_mesh_shortest_path::Shortest_path_result result = shortestPaths.shortest_distance_to_source_points(*currentVertex);
CHECK_CLOSE(result.first, expectedDistances2[i], Kernel::FT(0.0001));
assert(result.second == expectedSources2[i]);
++currentVertex;
}
// Test removing a source vertex
shortestPaths.remove_source_point(firstSourcePoint);
shortestPaths.build_sequence_tree();
// replace the only shortest path which was not reporting to the 2nd source point
expectedDistances2[1] = distanceToSaddle + shorterSideLength;
currentVertex = startVertex;
for (size_t i = 0; i < 8; ++i)
{
Surface_mesh_shortest_path::Shortest_path_result result = shortestPaths.shortest_distance_to_source_points(*currentVertex);
CHECK_CLOSE(result.first, expectedDistances2[i], Kernel::FT(0.0001));
assert(result.second == secondSourcePoint);
++currentVertex;
}
}
void test_boundary_mesh()
{
typedef CGAL::Exact_predicates_inexact_constructions_kernel Kernel;
typedef CGAL::Polyhedron_3<Kernel, CGAL::Polyhedron_items_with_id_3> Polyhedron_3;
typedef CGAL::Surface_mesh_shortest_path_traits<Kernel, Polyhedron_3> Traits;
typedef Traits::Barycentric_coordinate Barycentric_coordinate;
typedef Traits::FT FT;
typedef Traits::Point_3 Point_3;
typedef Traits::Triangle_3 Triangle_3;
typedef boost::graph_traits<Polyhedron_3> Graph_traits;
typedef Graph_traits::vertex_descriptor vertex_descriptor;
typedef Graph_traits::vertex_iterator vertex_iterator;
typedef Graph_traits::face_descriptor face_descriptor;
typedef Graph_traits::face_iterator face_iterator;
typedef CGAL::Surface_mesh_shortest_path<Traits> Surface_mesh_shortest_path;
typedef boost::property_map<Polyhedron_3, CGAL::vertex_point_t>::type VPM;
Traits traits;
Traits::Construct_triangle_3_to_triangle_2_projection project_triangle_3_to_triangle_2(traits.construct_triangle_3_to_triangle_2_projection_object());
CGAL_USE(project_triangle_3_to_triangle_2);
Traits::Compute_squared_distance_3 compute_squared_distance_3(traits.compute_squared_distance_3_object());
Traits::Construct_barycenter_3 construct_barycenter_3(traits.construct_barycenter_3_object());
Traits::Construct_triangle_3_along_segment_2_flattening flatten_triangle_3_along_segment_2(traits.construct_triangle_3_along_segment_2_flattening_object());
CGAL_USE(flatten_triangle_3_along_segment_2);
Traits::Construct_barycentric_coordinate construct_barycentric_coordinate(traits.construct_barycentric_coordinate_object());
struct Construct_barycenter_in_triangle_3
{
Traits::Construct_barycenter_3 m_cb3;
Construct_barycenter_in_triangle_3(Traits::Construct_barycenter_3 cb3)
: m_cb3(cb3)
{
}
Point_3 operator() (const Triangle_3& t, const Barycentric_coordinate& b)
{
return m_cb3(t[0], b[0], t[1], b[1], t[2], b[2]);
}
} construct_barycenter_in_triangle_3(construct_barycenter_3);
std::ifstream inFile("data/boundary_mesh.off");
Polyhedron_3 P;
inFile >> P;
inFile.close();
CGAL::set_halfedgeds_items_id(P);
face_iterator startFace;
face_iterator endFace;
boost::tie(startFace, endFace) = CGAL::faces(P);
vertex_iterator currentVertex;
vertex_iterator endVertex;
VPM vpm = CGAL::get(CGAL::vertex_point, P);
vertex_descriptor vertexHandles[10];
face_descriptor faceHandles[8];
Point_3 vertexLocations[10];
size_t currentVertexIndex = 0;
for (boost::tie(currentVertex, endVertex) = CGAL::vertices(P); currentVertex != endVertex; ++currentVertex)
{
vertexHandles[currentVertexIndex] = *currentVertex;
vertexLocations[currentVertexIndex] = vpm[*currentVertex];
++currentVertexIndex;
}
size_t currentFaceIndex = 0;
for (face_iterator currentFace = startFace; currentFace != endFace; ++currentFace)
{
faceHandles[currentFaceIndex] = *currentFace;
++currentFaceIndex;
}
Barycentric_coordinate startLocation = construct_barycentric_coordinate(FT(0.1), FT(0.8), FT(0.1));
typedef boost::property_map<Polyhedron_3, CGAL::face_external_index_t>::type FaceIndexMap;
FaceIndexMap faceIndexMap(CGAL::get(CGAL::face_external_index, P));
Surface_mesh_shortest_path shortestPaths(P, traits);
//shortestPaths.m_debugOutput = true;
shortestPaths.add_source_point(*startFace, startLocation);
shortestPaths.build_sequence_tree();
Triangle_3 firstTriangle(vertexLocations[1], vertexLocations[0], vertexLocations[2]);
Point_3 locationInTriangle(construct_barycenter_in_triangle_3(firstTriangle, startLocation));
FT dist0 = shortestPaths.shortest_distance_to_source_points(vertexHandles[0]).first;
CHECK_CLOSE(dist0, CGAL::sqrt(compute_squared_distance_3(locationInTriangle, vertexLocations[0])), FT(0.000001));
FT dist1 = shortestPaths.shortest_distance_to_source_points(vertexHandles[1]).first;
CHECK_CLOSE(dist1, CGAL::sqrt(compute_squared_distance_3(locationInTriangle, vertexLocations[1])), FT(0.000001));
FT dist2 = shortestPaths.shortest_distance_to_source_points(vertexHandles[2]).first;
CHECK_CLOSE(dist2, CGAL::sqrt(compute_squared_distance_3(locationInTriangle, vertexLocations[2])), FT(0.000001));
FT dist3 = shortestPaths.shortest_distance_to_source_points(vertexHandles[3]).first;
CHECK_CLOSE(dist3, CGAL::sqrt(compute_squared_distance_3(locationInTriangle, vertexLocations[3])), FT(0.000001));
FT dist4 = shortestPaths.shortest_distance_to_source_points(vertexHandles[4]).first;
CHECK_CLOSE(dist4, CGAL::sqrt(compute_squared_distance_3(locationInTriangle, vertexLocations[1])) + CGAL::sqrt(compute_squared_distance_3(vertexLocations[1], vertexLocations[4])), FT(0.000001));
FT dist5 = shortestPaths.shortest_distance_to_source_points(vertexHandles[5]).first;
CHECK_CLOSE(dist5, CGAL::sqrt(compute_squared_distance_3(locationInTriangle, vertexLocations[3])) + CGAL::sqrt(compute_squared_distance_3(vertexLocations[3], vertexLocations[5])), FT(0.000001));
Barycentric_coordinate somewhereElseInFirstTriangle = construct_barycentric_coordinate(0.8, 0.05, 0.15);
FT distT0 = shortestPaths.shortest_distance_to_source_points(faceHandles[0], somewhereElseInFirstTriangle).first;
CHECK_CLOSE(distT0, CGAL::sqrt(compute_squared_distance_3(locationInTriangle, construct_barycenter_in_triangle_3(firstTriangle, somewhereElseInFirstTriangle))), FT(0.000001));
Triangle_3 oneStepTriangle(vertexLocations[4], vertexLocations[1], vertexLocations[3]);
Barycentric_coordinate locationInOneStepTriangle = construct_barycentric_coordinate(0.1, 0.8, 0.1);
CGAL::test::Edge_sequence_collector<Traits> collector(P);
shortestPaths.shortest_path_sequence_to_source_points(faceHandles[2], locationInOneStepTriangle, collector);
FT distT2 = shortestPaths.shortest_distance_to_source_points(faceHandles[2], locationInOneStepTriangle).first;
CHECK_CLOSE(distT2, dist1 + CGAL::sqrt(compute_squared_distance_3(vertexLocations[1], construct_barycenter_in_triangle_3(oneStepTriangle, locationInOneStepTriangle))), FT(0.00001));
Triangle_3 twoStepTriangle(vertexLocations[6], vertexLocations[5], vertexLocations[7]);
Barycentric_coordinate locationInTwoStepTriangle = construct_barycentric_coordinate(0.8, 0.1, 0.1);
FT distT5 = shortestPaths.shortest_distance_to_source_points(faceHandles[5], locationInTwoStepTriangle).first;
CHECK_CLOSE(distT5, dist3 + CGAL::sqrt(compute_squared_distance_3(vertexLocations[3], construct_barycenter_in_triangle_3(twoStepTriangle, locationInTwoStepTriangle))), FT(0.00001));
Triangle_3 threeStepTriangle(vertexLocations[7], vertexLocations[5], vertexLocations[8]);
Barycentric_coordinate locationInThreeStepTriangle = construct_barycentric_coordinate(0.2, 0.6, 0.2);
FT distT6 = shortestPaths.shortest_distance_to_source_points(faceHandles[6], locationInThreeStepTriangle).first;
CHECK_CLOSE(distT6, dist5 + CGAL::sqrt(compute_squared_distance_3(vertexLocations[5], construct_barycenter_in_triangle_3(threeStepTriangle, locationInThreeStepTriangle))), FT(0.00001));
}
