How to estimate the projected area of a boost::polygon C++

Question

I need to estimate the projected area of a boost::polygon of several circles that are created in a loop.

The idea is for a given x and y centroid, a polygon is created drawing a circle of a given radius using n number of vertices. This circle is stored in a ‘polygon_90_set_data’. These circles may or not overlap between them. This is repeated several times in a loop with the idea to create all the circles of interest (> 2 mill). I need to estimate the area projected by all circles similar to the enclosed figure.

However, when I try to get the projected area in my code I just get the area of a circle and not all of them. This could be a problem of dissolve or merge, but not sure how do it. Could you help me? This is my code so far.

#include <vector>
#include <boost/polygon/polygon.hpp>
#include "draw_circle_rcpp.h"

namespace bpl = boost::polygon;
typedef bpl::polygon_90_data<double> Polygon;
typedef bpl::polygon_traits<Polygon>::point_type Point;
typedef bpl::polygon_90_set_data<double> PolygonSet;

using namespace boost::polygon::operators;

double displayed_area_rcpp(arma::mat spheres, int nvertices = 10L) {

  const std::size_t n_shperes = spheres.n_rows;

  // Create a polygon
  PolygonSet da;

  for (int i = 0; i < n_shperes; i++) {

    // Create a polygon
    Polygon polygon;

    double X = spheres(i, 0);
    double Y = spheres(i, 1);
    double radius = spheres(i, 2);
    
    //Create the circle vertices
    arma::mat circle_sphere = draw_circle_rcpp(X, Y, radius, nvertices);

    // Create points
    std::vector<Point> points;

    for (int j = 0; j < nvertices; ++j) {

      double xx = circle_sphere(j, 0);
      double yy = circle_sphere(j, 1);

      points.push_back(Point(xx, yy));

    }

    // Points to polygon
    polygon.set(points.begin(),points.end());

    assign(da, polygon);
  }

  return bpl::area(da);
} 

I think that the potential issue is where I do the 'assign' but not sure.