c++ - Boost::Geometry - 在 3d 空间中查找 2d 多边形的面积？

Question

我正在尝试获取 3d 空间中 2d 多边形的面积。有没有办法通过 Boost::Geometry 做到这一点？这是我的实现，但它始终返回 0:

#include <iostream>

#include <boost/geometry.hpp>
#include <boost/geometry/geometries/point_xy.hpp>
#include <boost/geometry/geometries/polygon.hpp>
#include <boost/geometry/io/wkt/wkt.hpp>

namespace bg = boost::geometry;

typedef bg::model::point<double, 3, bg::cs::cartesian> point3d;

int main()
{
    bg::model::multi_point<point3d> square;
    bg::read_wkt("MULTIPOINT((0 0 0), (0 2 0), (0 2 2), (0 0 2), (0 0 0))", square);
    double area = bg::area(square);
    std::cout << "Area: " << area << std::endl;

    return 0;
}

UPD:实际上，我对简单的 2d 多点正方形有同样的问题:

#include <iostream>

#include <boost/geometry.hpp>
#include <boost/geometry/geometries/point_xy.hpp>
#include <boost/geometry/geometries/polygon.hpp>
#include <boost/geometry/io/wkt/wkt.hpp>

namespace bg = boost::geometry;

typedef bg::model::point<double, 2, bg::cs::cartesian> point2d;

int main()
{
    bg::model::multi_point<point2d> square;
    bg::read_wkt("MULTIPOINT((0 0), (2 0), (2 2), (0 2))", square);
    double area = bg::area(square);
    std::cout << "Area: " << area << std::endl;

    return 0;
}

结果如下:

$ ./test_area
Area: 0

UPD:看起来像 boost::geometry 中的面积计算仅适用于二维多边形。

Answer 1

我不希望点的集合有一个区域。你需要相当于一个 model::polygon<poind3d>但目前似乎不受支持。

如果保证点共面并且线段彼此不相交，您可以将多边形分解为一系列三角形，并根据以下公式用一点线性代数计算面积对于三角形的面积:

在非凸多边形的情况下，需要调整面积之和以减去多边形以外的区域。实现这一点的最简单方法是使用三角形的有符号区域，包括右手三角形的正贡献和左手三角形的负贡献:

请注意，似乎有一些计划包括 cross_product Boost 中的实现，但从 1.56 版开始它似乎并未包含在内。以下替换应该可以解决您的用例问题:

point3d cross_product(const point3d& p1, const point3d& p2)
{
  double x = bg::get<0>(p1);
  double y = bg::get<1>(p1);
  double z = bg::get<2>(p1);
  double u = bg::get<0>(p2);
  double v = bg::get<1>(p2);
  double w = bg::get<2>(p2);
  return point3d(y*w-z*v, z*u-x*w, x*v-y*u);
}
point3d cross_product(const bg::model::segment<point3d>& p1
                    , const bg::model::segment<point3d>& p2)
{
  point3d v1(p1.second);
  point3d v2(p2.second);
  bg::subtract_point(v1, p1.first);
  bg::subtract_point(v2, p2.first);

  return cross_product(v1, v2);
}

然后可以用以下方法计算面积:

// compute the are of a collection of 3D points interpreted as a 3D polygon
// Note that there are no checks as to whether or not the points are
// indeed co-planar.
double area(bg::model::multi_point<point3d>& polygon)
{
  if (polygon.size()<3) return 0;

  bg::model::segment<point3d> v1(polygon[1], polygon[0]);
  bg::model::segment<point3d> v2(polygon[2], polygon[0]);
  // Compute the cross product for the first pair of points, to handle
  // shapes that are not convex.
  point3d n1 = cross_product(v1, v2);
  double normSquared = bg::dot_product(n1, n1);
  if (normSquared > 0)
  {
    bg::multiply_value(n1, 1.0/sqrt(normSquared));
  }
  // sum signed areas of triangles
  double result = 0.0;
  for (size_t i=1; i<polygon.size(); ++i)
  {
    bg::model::segment<point3d> v1(polygon[0], polygon[i-1]);
    bg::model::segment<point3d> v2(polygon[0], polygon[i]);

    result += bg::dot_product(cross_product(v1, v2), n1);
  }
  result *= 0.5;
  return abs(result);
}