【CGAL_网格】Surface

官方文档链接：CGAL 5.4.2 – Surface Mesh: User Manual

0 概述

Surface_mesh 类是半边数据结构的实现，可用于表示多面体表面。

相较于 Halfedge Data Structures 和 3D Polyhedral Surface ，其具有以下特点：

不同于基于指针，Surface_mesh基于索引（数据结构使用整数索引作为顶点、半边、边和面的描述符）；
向顶点、半边、边和面添加信息的机制要简单得多；
具有更低的内存占用。
元素被移除后，会被标记为已移除，需要调用垃圾回收函数才能实现真正地移除。

1 用法

Surface_mesh类提供了如下四个类来表示网格的基本元素：

Surface_mesh::Vertex_index
Surface_mesh::Halfedge_index
Surface_mesh::Face_index
Surface_mesh::Edge_index

由于Surface_mesh是基于索引值构造的，所以没有访问连接性或属性的成员函数。

1.1 栗子

#include <CGAL/Simple_cartesian.h>
#include <CGAL/Surface_mesh.h>
typedef CGAL::Simple_cartesian<double> K;
typedef CGAL::Surface_mesh<K::Point_3> Mesh;
typedef Mesh::Vertex_index vertex_descriptor;   //基于索引的顶点描述
typedef Mesh::Face_index face_descriptor;       //基于索引的面描述
int main()
{
    Mesh m;
    // Add the points as vertices
    vertex_descriptor u = m.add_vertex(K::Point_3(0, 1, 0));
    vertex_descriptor v = m.add_vertex(K::Point_3(0, 0, 0));
    vertex_descriptor w = m.add_vertex(K::Point_3(1, 1, 0));
    vertex_descriptor x = m.add_vertex(K::Point_3(1, 0, 0));
    m.add_face(u, v, w);
    face_descriptor f = m.add_face(u, v, x);
    if (f == Mesh::null_face())
    {
        std::cerr << "The face could not be added because of an orientation error." << std::endl;
        f = m.add_face(u, x, v);
        assert(f != Mesh::null_face());
    }
    return 0;
}

在这个例子中，通过添加两个面创建一个简单的表面网格。可以看到，在使用add_face()添加face的时候，若其返回的Face_index值为Surface_mesh::null_face()则说明添加面的操作在拓扑上无效，添加失败。

2 连接关系

在一个surface_mesh中可以使用如下函数进行基于连接关系的查询：
Surface_mesh::opposite(), Surface_mesh::next(), Surface_mesh::prev(), Surface_mesh::target(), and Surface_mesh::face()。此外，函数Surface_mesh::halfedge() 能够获取与顶点和面相关联的半边。

范围和迭代器

surface_mesh提供了迭代器用以枚举所有的顶点、半边、边和面。

#include <vector>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Surface_mesh.h>
typedef CGAL::Simple_cartesian<double> K;
typedef CGAL::Surface_mesh<K::Point_3> Mesh;
typedef Mesh::Vertex_index vertex_descriptor;
typedef Mesh::Face_index face_descriptor;
int main()
{
  Mesh m;
  // u            x
  // +------------+
  // |            |
  // |            |
  // |      f     |
  // |            |
  // |            |
  // +------------+
  // v            w
  // Add the points as vertices
  vertex_descriptor u = m.add_vertex(K::Point_3(0,1,0));
  vertex_descriptor v = m.add_vertex(K::Point_3(0,0,0));
  vertex_descriptor w = m.add_vertex(K::Point_3(1,0,0));
  vertex_descriptor x = m.add_vertex(K::Point_3(1,1,0));
  /* face_descriptor f = */ m.add_face(u,v,w,x);
  {
    std::cout << "all vertices " << std::endl;
    // The vertex iterator type is a nested type of the Vertex_range
    Mesh::Vertex_range::iterator  vb, ve;
    Mesh::Vertex_range r = m.vertices();
    // The iterators can be accessed through the C++ range API
    vb = r.begin();
    ve = r.end();
    // or the boost Range API
    vb = boost::begin(r);
    ve = boost::end(r);
    // or with boost::tie, as the CGAL range derives from std::pair
    for(boost::tie(vb, ve) = m.vertices(); vb != ve; ++vb){
            std::cout << *vb << std::endl;
    }
    // Instead of the classical for loop one can use
    // the boost macro for a range
    for(vertex_descriptor vd : m.vertices()){
      std::cout << vd << std::endl;
    }
    // or the C++11 for loop. Note that there is a ':' and not a ',' as in BOOST_FOREACH
    for(vertex_descriptor vd : m.vertices()){
      std::cout << vd << std::endl;
    }
  }
  return 0;
}

这是一个遍历表面网格中所有顶点索引的例子。其中关键迭代器类型Mesh::Vertex_range::iterator，关键范围类型Mesh::Vertex_range。通过后者得到前者，进而对范围中元素进行迭代遍历。

4 循环器

围绕面的循环器：

CGAL::Halfedge_around_face_circulator<Mesh>
CGAL::Vertex_around_face_circulator<Mesh>
CGAL::Face_around_face_circulator<Mesh>

围绕目标顶点的循环器：

CGAL::Halfedge_around_target_circulator<Mesh>
CGAL::Vertex_around_target_circulator<Mesh>
CGAL::Face_around_target_circulator<Mesh>

所有循环器围绕方向为逆时针。

#include <CGAL/Simple_cartesian.h>
#include <CGAL/Surface_mesh.h>
#include <vector>
typedef CGAL::Simple_cartesian<double> K;
typedef CGAL::Surface_mesh<K::Point_3> Mesh;
typedef Mesh::Vertex_index vertex_descriptor;
typedef Mesh::Face_index face_descriptor;
int main()
{
  Mesh m;
  // u            x
  // +------------+
  // |            |
  // |            |
  // |      f     |
  // |            |
  // |            |
  // +------------+
  // v            w
  // Add the points as vertices
  vertex_descriptor u = m.add_vertex(K::Point_3(0,1,0));
  vertex_descriptor v = m.add_vertex(K::Point_3(0,0,0));
  vertex_descriptor w = m.add_vertex(K::Point_3(1,0,0));
  vertex_descriptor x = m.add_vertex(K::Point_3(1,1,0));
  face_descriptor f = m.add_face(u,v,w,x);
  {
    std::cout << "vertices around vertex " << v << std::endl;
    CGAL::Vertex_around_target_circulator<Mesh> vbegin(m.halfedge(v),m), done(vbegin);
    do {
      std::cout << *vbegin++ << std::endl;
    } while(vbegin != done);
  }
  {
    std::cout << "vertices around face " << f << std::endl;
    CGAL::Vertex_around_face_iterator<Mesh> vbegin, vend;
    for(boost::tie(vbegin, vend) = vertices_around_face(m.halfedge(f), m);
        vbegin != vend;
        ++vbegin){
      std::cout << *vbegin << std::endl;
    }
  }
  // or the same again, but directly with a range based loop
  for(vertex_descriptor vd : vertices_around_face(m.halfedge(f), m)){
    std::cout << vd << std::endl;
  }
  return 0;
}

这个例子中展示了Vertex_around_target_circulator的使用方法，其中也展示了使用迭代器达到相同的效果。

5 属性

surface_mesh提供了可以在运行时指定顶点、半边、边和面的属性的机制。每个属性通过一个字符串和它的键类型进行标识。

默认情况下，只有一个属性"v:point"。当通过 Surface_mesh::add_vertex() 向数据结构添加新点时，必须提供此属性的值。可以使用 Surface_mesh::points() 或 Surface_mesh::point(Surface_mesh::Vertex_index v) 直接访问该属性。

当一个元素被移除，它并不会立刻被真正意义上的删除，而是被标记为“已移除”，只有调用Surface_mesh::collect_garbage()才能真正地移除该元素的所有相关属性。

#include <string>
#include <CGAL/Simple_cartesian.h>
#include <CGAL/Surface_mesh.h>
typedef CGAL::Simple_cartesian<double> K;
typedef CGAL::Surface_mesh<K::Point_3> Mesh;
typedef Mesh::Vertex_index vertex_descriptor;
typedef Mesh::Face_index face_descriptor;
int main()
{
  Mesh m;
  vertex_descriptor v0 = m.add_vertex(K::Point_3(0,2,0));
  vertex_descriptor v1 = m.add_vertex(K::Point_3(2,2,0));
  vertex_descriptor v2 = m.add_vertex(K::Point_3(0,0,0));
  vertex_descriptor v3 = m.add_vertex(K::Point_3(2,0,0));
  vertex_descriptor v4 = m.add_vertex(K::Point_3(1,1,0));
  m.add_face(v3, v1, v4);
  m.add_face(v0, v4, v1);
  m.add_face(v0, v2, v4);
  m.add_face(v2, v3, v4);
  // give each vertex a name, the default is empty
  Mesh::Property_map<vertex_descriptor,std::string> name;
  bool created;
  boost::tie(name, created) = m.add_property_map<vertex_descriptor,std::string>("v:name","");
  assert(created);
  // add some names to the vertices
  name[v0] = "hello";
  name[v2] = "world";
  {
    // You get an existing property, and created will be false
    Mesh::Property_map<vertex_descriptor,std::string> name;
    bool created;
    boost::tie(name, created) = m.add_property_map<vertex_descriptor,std::string>("v:name", "");
    assert(! created);
  }
  //  You can't get a property that does not exist
  Mesh::Property_map<face_descriptor,std::string> gnus;
  bool found;
  boost::tie(gnus, found) = m.property_map<face_descriptor,std::string>("v:gnus");
  assert(! found);
  // retrieve the point property for which exists a convenience function
  Mesh::Property_map<vertex_descriptor, K::Point_3> location = m.points();
  for(vertex_descriptor vd : m.vertices()) {
    std::cout << name[vd] << " @ " << location[vd] << std::endl;
  }
  std::vector<std::string> props = m.properties<vertex_descriptor>();
  for(std::string p : props){
    std::cout << p << std::endl;
  }
  // delete the string property again
  m.remove_property_map(name);
  return 0;
}

上面例子中依次实现了property的创建、查询、删除等操作。

6 边界

一个半边存储了其入射面的引用，如果它没有入射面，则表示该半边在边界上，即sm.face(h) == Surface_mesh::null_face()。若一个半边在边界上，那么包含它的边和以它为入射半边的顶点都位于边界上。

Surface_mesh::is_border(Vertex_index v, bool check_all_incident_halfedges = false)：检查一个顶点的关联半边是否在边界上。

Surface_mesh::set_vertex_halfedge_to_border_halfedge(Vertex_index v) //将单个顶点的关联半边设置为边界半边
Surface_mesh::set_vertex_halfedge_to_border_halfedge(Halfedge_index h) //将面h的边界顶点的关联半边设置为边界半边
Surface_mesh::set_vertex_halfedge_to_border_halfedge() //将单表面网格上所有顶点的关联半边设置为边界半边

7 相关API

7.1 栗子

#include <CGAL/Simple_cartesian.h>
#include <CGAL/Surface_mesh.h>
#include <boost/graph/kruskal_min_spanning_tree.hpp>
#include <iostream>
#include <fstream>
#include <list>
typedef CGAL::Simple_cartesian<double>                       Kernel;
typedef Kernel::Point_3                                      Point;
typedef CGAL::Surface_mesh<Point>                            Mesh;
typedef boost::graph_traits<Mesh>::vertex_descriptor vertex_descriptor;
typedef boost::graph_traits<Mesh>::vertex_iterator   vertex_iterator;
typedef boost::graph_traits<Mesh>::edge_descriptor   edge_descriptor;
void kruskal(const Mesh& sm)
{
   // We use the default edge weight which is the squared length of the edge
  std::list<edge_descriptor> mst;
  boost::kruskal_minimum_spanning_tree(sm,
                                       std::back_inserter(mst));
  std::cout << "#VRML V2.0 utf8\\n"
    "Shape {\\n"
    "  appearance Appearance {\\n"
    "    material Material { emissiveColor 1 0 0}}\\n"
    "    geometry\\n"
    "    IndexedLineSet {\\n"
    "      coord Coordinate {\\n"
    "        point [ \\n";
  vertex_iterator vb,ve;
  for(boost::tie(vb, ve) = vertices(sm); vb!=ve; ++vb){
    std::cout <<  "        " << sm.point(*vb) << "\\n";
  }
  std::cout << "        ]\\n"
               "     }\\n"
    "      coordIndex [\\n";
  for(std::list<edge_descriptor>::iterator it = mst.begin(); it != mst.end(); ++it)
  {
    edge_descriptor e = *it ;
    vertex_descriptor s = source(e,sm);
    vertex_descriptor t = target(e,sm);
    std::cout << "      " << s << ", " << t <<  ", -1\\n";
  }
  std::cout << "]\\n"
    "  }#IndexedLineSet\\n"
    "}# Shape\\n";
}
int main(int argc, char** argv)
{
  Mesh sm;
  std::string fname = argc==1?CGAL::data_file_path("meshes/knot1.off"):argv[1];
  if(!CGAL::IO::read_polygon_mesh(fname, sm))
  {
    std::cerr << "Invalid input file." << std::endl;
    return EXIT_FAILURE;
  }
  kruskal(sm);
  return 0;
}