// Copyright (c) 2005  INRIA Sophia-Antipolis (France).
// All rights reserved.
//
// This file is part of CGAL (www.cgal.org); you may redistribute it under
// the terms of the Q Public License version 1.0.
// See the file LICENSE.QPL distributed with CGAL.
//
// Licensees holding a valid commercial license may use this file in
// accordance with the commercial license agreement provided with the software.
//
// This file is provided AS IS with NO WARRANTY OF ANY KIND, INCLUDING THE
// WARRANTY OF DESIGN, MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE.
//
// $URL: svn+ssh://scm.gforge.inria.fr/svn/cgal/trunk/Principal_component_analysis/include/CGAL/linear_least_squares_fitting_triangles_2.h $
// $Id: linear_least_squares_fitting_triangles_2.h 56667 2010-06-09 07:37:13Z sloriot $
//
// Author(s) : Pierre Alliez and Sylvain Pion and Ankit Gupta

#ifndef CGAL_LINEAR_LEAST_SQUARES_FITTING_TRIANGLES_2_H
#define CGAL_LINEAR_LEAST_SQUARES_FITTING_TRIANGLES_2_H

#include <CGAL/basic.h>
#include <CGAL/Object.h>
#include <CGAL/centroid.h>
#include <CGAL/eigen_2.h>
#include <CGAL/eigen.h>
#include <CGAL/Linear_algebraCd.h>
#include <CGAL/PCA_util.h>

#include <iterator>
#include <vector>
#include <cmath>
#include <list>

namespace CGAL {

namespace internal {
// Fits a line to a 2D triangle set.
// Returns a fitting quality (1 - lambda_min/lambda_max):
//  1 is best  (zero variance orthogonally to the fitting line);
//  0 is worst (isotropic case, returns a line with horizontal
//              direction by default)

template < typename InputIterator,
           typename Kernel >
typename Kernel::FT
linear_least_squares_fitting_2(InputIterator first,
                               InputIterator beyond, 
                               typename Kernel::Line_2& line,   // best fit line
                               typename Kernel::Point_2& c,     // centroid
                               const typename Kernel::Triangle_2*,// used for indirection
                               const Kernel&,                   // kernel
			                         const CGAL::Dimension_tag<2>& tag)
{
  // types
  typedef typename Kernel::FT       FT;
  typedef typename Kernel::Line_2   Line;
  typedef typename Kernel::Point_2  Point;
  typedef typename Kernel::Vector_2 Vector;
  typedef typename Kernel::Triangle_2 Triangle;
  typedef typename CGAL::Linear_algebraCd<FT> LA;
  typedef typename LA::Matrix Matrix;

  // precondition: at least one element in the container.
  CGAL_precondition(first != beyond);

  // compute centroid
  c = centroid(first,beyond,Kernel(),tag);

  // assemble covariance matrix as a semi-definite matrix. 
  // Matrix numbering:
  // 0
  // 1 2
  //Final combined covariance matrix for all triangles and their combined mass
  FT mass = 0.0;
  FT covariance[3] = {0.0, 0.0, 0.0};

  // assemble the 2nd order moment about the origin.  
  FT temp[4] = {1/12.0, 1/24.0,
		1/24.0, 1/12.0};

  Matrix moment = init_matrix<Kernel>(2,temp);

  for(InputIterator it = first;
      it != beyond;
      it++)
  {
    // Now for each triangle, construct the 2nd order moment about the origin.
    // assemble the transformation matrix.
    const Triangle& t = *it;

    // defined for convenience.
    FT x0 = t[0].x();
    FT y0 = t[0].y();
    FT delta[4] = {t[1].x() - x0, t[2].x() - x0,
		   t[1].y() - y0, t[2].y() - y0};

    Matrix transformation = init_matrix<Kernel>(2,delta);
    FT area = 0.5 * std::abs(LA::determinant(transformation));
    CGAL_assertion(area!=0);

    // Find the 2nd order moment for the triangle wrt to the origin by an affine transformation.
    
    // Transform the standard 2nd order moment using the transformation matrix
    transformation = 2 * area * transformation * moment * LA::transpose(transformation);
    
    // Translate the 2nd order moment to (x0,y0).
    FT xav0 = (delta[0]+delta[1])/3.0;
    FT yav0 = (delta[2]+delta[3])/3.0;
    
    // and add to the covariance matrix
    covariance[0] += transformation[0][0] + area * (x0*xav0*2 + x0*x0);
    covariance[1] += transformation[0][1] + area * (x0*yav0 + xav0*y0 + x0*y0);
    covariance[2] += transformation[1][1] + area * (y0*yav0*2 + y0*y0);

    mass += area;
  }
  
  // Translate the 2nd order moment calculated about the origin to 
  // the center of mass to get the covariance.
  covariance[0] += mass * (-1.0 * c.x() * c.x());
  covariance[1] += mass * (-1.0 * c.x() * c.y());
  covariance[2] += mass * (-1.0 * c.y() * c.y());

  //  std::cout<<"cov: "<<covariance[0]*covariance[2]<<" =? "<<covariance[1]*covariance[1]<<std::endl;

  // solve for eigenvalues and eigenvectors.
  // eigen values are sorted in descending order, 
  // eigen vectors are sorted in accordance.
  std::pair<FT,FT> eigen_values;
  std::pair<Vector,Vector> eigen_vectors;
  FT eigen_vectors1[4];
  FT eigen_values1[2];
  eigen_symmetric<FT>(covariance,2, eigen_vectors1, eigen_values1);
  eigen_values = std::make_pair(eigen_values1[0],eigen_values1[1]);
  eigen_vectors = std::make_pair(Vector(eigen_vectors1[0],eigen_vectors1[1]),Vector(eigen_vectors1[2],eigen_vectors1[3]));
  // check unicity and build fitting line accordingly
  if(eigen_values.first != eigen_values.second)
  {
    // regular case
    line = Line(c, eigen_vectors.first);
    return (FT)1.0 - eigen_values.second / eigen_values.first;
  } 
  else
  {
    // isotropic case (infinite number of directions)
    // by default: assemble a line that goes through 
    // the centroid and with a default horizontal vector.
    line = Line(c, Vector(1.0, 0.0));
    return (FT)0.0;
  } 
} // end linear_least_squares_fitting_2 for triangle set with 2D tag

template < typename InputIterator,
           typename Kernel >
typename Kernel::FT
linear_least_squares_fitting_2(InputIterator first,
                               InputIterator beyond, 
                               typename Kernel::Line_2& line,   // best fit line
                               typename Kernel::Point_2& c,     // centroid
                               const typename Kernel::Triangle_2*,// used for indirection
                               const Kernel&,                   // kernel
			                         const CGAL::Dimension_tag<1>& tag)
{
  // types
  typedef typename Kernel::Triangle_2 Triangle;
  typedef typename Kernel::Segment_2  Segment;

  // precondition: at least one element in the container.
  CGAL_precondition(first != beyond);
  
  std::list<Segment> segments;  
  for(InputIterator it = first;
      it != beyond;
      it++)
  {
    const Triangle& t = *it;
    segments.push_back(Segment(t[0],t[1]));
    segments.push_back(Segment(t[1],t[2]));
    segments.push_back(Segment(t[2],t[0]));      
  }    
  
  return linear_least_squares_fitting_2(segments.begin(),segments.end(),line,c,tag,Kernel());
  
} // end linear_least_squares_fitting_2 for triangle set with 1D tag

template < typename InputIterator,
           typename Kernel >
typename Kernel::FT
linear_least_squares_fitting_2(InputIterator first,
                               InputIterator beyond, 
                               typename Kernel::Line_2& line,   // best fit line
                               typename Kernel::Point_2& c,     // centroid
                               const typename Kernel::Triangle_2*,// used for indirection
                               const Kernel&,                   // kernel
			                         const CGAL::Dimension_tag<0>& tag)
{
  // types

  typedef typename Kernel::Triangle_2 Triangle;
  typedef typename Kernel::Point_2 Point;

  // precondition: at least one element in the container.
  CGAL_precondition(first != beyond);
  
  std::list<Point> points;  
  for(InputIterator it = first;
      it != beyond;
      it++)
  {
    const Triangle& t = *it;
    points.push_back(Point(t[0]));
    points.push_back(Point(t[1]));
    points.push_back(Point(t[2]));      
  }    
  
  return linear_least_squares_fitting_2(points.begin(),points.end(),line,c,tag,Kernel());
  
} // end linear_least_squares_fitting_2 for triangle set with 0D tag

} // end namespace internal

} //namespace CGAL

#endif // CGAL_LINEAR_LEAST_SQUARES_FITTING_TRIANGLES_2_H
