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 * $Id: intersections.cpp 6126 2012-07-03 20:19:58Z aichim $
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#include <pcl/common/intersections.h>

bool
pcl::lineWithLineIntersection (const Eigen::VectorXf &line_a, 
                               const Eigen::VectorXf &line_b, 
                               Eigen::Vector4f &point, double sqr_eps)
{
  Eigen::Vector4f p1, p2;
  lineToLineSegment (line_a, line_b, p1, p2);

  // If the segment size is smaller than a pre-given epsilon...
  double sqr_dist = (p1 - p2).squaredNorm ();
  if (sqr_dist < sqr_eps)
  {
    point = p1;
    return (true);
  }
  point.setZero ();
  return (false);
}

bool
pcl::lineWithLineIntersection (const pcl::ModelCoefficients &line_a, 
                               const pcl::ModelCoefficients &line_b, 
                               Eigen::Vector4f &point, double sqr_eps)
{
  Eigen::VectorXf coeff1 = Eigen::VectorXf::Map (&line_a.values[0], line_a.values.size ());
  Eigen::VectorXf coeff2 = Eigen::VectorXf::Map (&line_b.values[0], line_b.values.size ());
  return (lineWithLineIntersection (coeff1, coeff2, point, sqr_eps));
}

bool 
pcl::planeWithPlaneIntersection (const Eigen::Vector4f &plane_a, 
                                 const Eigen::Vector4f &plane_b,
                                 Eigen::VectorXf &line,
                                 double angular_tolerance)
{
  //planes shouldn't be parallel
  double test_cosine = plane_a.head<3>().dot(plane_b.head<3>());
  double upper_limit = 1 + angular_tolerance;
  double lower_limit = 1 - angular_tolerance;

  if ((test_cosine < upper_limit) && (test_cosine > lower_limit))
  {
      PCL_ERROR ("Plane A and Plane B are Parallel");
      return (false);
  }

  if ((test_cosine > -upper_limit) && (test_cosine < -lower_limit))
  {
      PCL_ERROR ("Plane A and Plane B are Parallel");
      return (false);
  }

  Eigen::Vector4f line_direction = plane_a.cross3(plane_b);
  line_direction.normalized();

  //construct system of equations using lagrange multipliers with one objective function and two constraints
  Eigen::MatrixXf langegrange_coefs(5,5);
  langegrange_coefs << 2,0,0,plane_a[0],plane_b[0],  0,2,0,plane_a[1],plane_b[1],  0,0,2, plane_a[2], plane_b[2], plane_a[0], plane_a[1] , plane_a[2], 0,0, plane_b[0], plane_b[1], plane_b[2], 0,0;

  Eigen::VectorXf b;
  b.resize(5);
  b << 0, 0, 0, -plane_a[3], -plane_b[3];

  //solve for the lagrange Multipliers
  Eigen::VectorXf x;
  x.resize(5);
  x = langegrange_coefs.colPivHouseholderQr().solve(b);

  line.resize(6);
  line.head<3>() = x.head<3>(); // the x[3] and x[4] are the values of the lagrange multipliers and are neglected
  line[3] = line_direction[0];
  line[4] = line_direction[1];
  line[5] = line_direction[2];
  return true;
} 
