libigl_Martin/include/igl/mvc.h

//
//  mvc.h
//
//  Created by Olga Diamanti on 9/11/11.
//  Copyright (c) 2011 ETH Zurich. All rights reserved.
//

#ifndef IGL_MVC_H
#define IGL_MVC_H

#include <Eigen/Core>

namespace igl 
{
//   MVC - MEAN VALUE COORDINATES
//  
//   mvc(V,C,W)
//  
//   Inputs:
//    V  #V x dim list of vertex positions (dim = 2 or dim = 3)
//    C  #C x dim list of polygon vertex positions in counter-clockwise order  (dim = 2 or dim = 3)
//  
//   Outputs:
//    W  weights, #V by #C matrix of weights
//  
    
  inline void mvc(const Eigen::MatrixXd &V, const Eigen::MatrixXd &C, Eigen::MatrixXd &W);
  
}

// Broken Implementation
inline void igl::mvc(const Eigen::MatrixXd &V, const Eigen::MatrixXd &C, Eigen::MatrixXd &W)
{
  
  // at least three control points
  assert(C.rows()>2);
  
  // dimension of points
  assert(C.cols() == 3 || C.cols() == 2);
  assert(V.cols() == 3 || V.cols() == 2);
  
  // number of polygon points
  int num = C.rows();
  
  Eigen::MatrixXd V1,C1;
  int i_prev, i_next;
  
  // check if either are 3D but really all z's are 0
  bool V_flat = (V.cols() == 3) && (std::sqrt( (V.col(3)).dot(V.col(3)) ) < 1e-10);
  bool C_flat = (C.cols() == 3) && (std::sqrt( (C.col(3)).dot(C.col(3)) ) < 1e-10);

  // if both are essentially 2D then ignore z-coords
  if((C.cols() == 2 || C_flat) && (V.cols() == 2 || V_flat))
  {
    // ignore z coordinate
    V1 = V.block(0,0,V.rows(),2);
    C1 = C.block(0,0,C.rows(),2);
  }
  else
  {
    // give dummy z coordinate to either mesh or poly
    if(V.rows() == 2)
    {
      V1 = Eigen::MatrixXd(V.rows(),3);
      V1.block(0,0,V.rows(),2) = V;
    }
    else
      V1 = V;

    if(C.rows() == 2)
    {
      C1 = Eigen::MatrixXd(C.rows(),3);
      C1.block(0,0,C.rows(),2) = C;
    }
    else
      C1 = C;

    // check that C is planar
    // average normal around poly corners

    Eigen::Vector3d n = Eigen::Vector3d::Zero();
    // take centroid as point on plane
    Eigen::Vector3d p = Eigen::Vector3d::Zero();
    for (int i = 0; i<num; ++i)
    {
      i_prev = (i>0)?(i-1):(num-1);
      i_next = (i<num-1)?(i+1):0;
      Eigen::Vector3d vnext = (C1.row(i_next) - C1.row(i)).transpose();
      Eigen::Vector3d vprev = (C1.row(i_prev) - C1.row(i)).transpose();
      n += vnext.cross(vprev);
      p += C1.row(i);
    }
    p/=num;
    n/=num;
    // normalize n
    n /= std::sqrt(n.dot(n));
    
    // check that poly is really coplanar
    for (int i = 0; i<num; ++i)
    {
      double dist_to_plane_C = std::abs((C1.row(i)-p.transpose()).dot(n));
      assert(dist_to_plane_C<1e-10);
    }
    
    // check that poly is really coplanar
    for (int i = 0; i<V1.rows(); ++i)
    {
      double dist_to_plane_V = std::abs((V1.row(i)-p.transpose()).dot(n));
      if(dist_to_plane_V>1e-10)
        std::cerr<<"Distance from V to plane of C is large..."<<std::endl;
    }
    
    // change of basis
    Eigen::Vector3d b1 = C1.row(1)-C1.row(0);
    Eigen::Vector3d b2 = n.cross(b1);
    // normalize basis rows
    b1 /= std::sqrt(b1.dot(b1));
    b2 /= std::sqrt(b2.dot(b2));
    n /= std::sqrt(n.dot(n));
    
    //transpose of the basis matrix in the m-file
    Eigen::Matrix3d basis = Eigen::Matrix3d::Zero();
    basis.col(0) = b1;
    basis.col(1) = b2;
    basis.col(2) = n;
    
    // change basis of rows vectors by right multiplying with inverse of matrix
    // with basis vectors as rows
    Eigen::ColPivHouseholderQR<Eigen::Matrix3d> solver = basis.colPivHouseholderQr();
    // Throw away coordinates in normal direction
    V1 = solver.solve(V1.transpose()).transpose().block(0,0,V1.rows(),2);
    C1 = solver.solve(C1.transpose()).transpose().block(0,0,C1.rows(),2);
    
  }
  
  // vectors from V to every C, where CmV(i,j,:) is the vector from domain
  // vertex j to handle i
  double EPSILON = 1e-10;
  Eigen::MatrixXd WW = Eigen::MatrixXd(C1.rows(), V1.rows());
  Eigen::MatrixXd dist_C_V (C1.rows(), V1.rows());
  std::vector< std::pair<int,int> > on_corner(0);
  std::vector< std::pair<int,int> > on_segment(0);
  for (int i = 0; i<C1.rows(); ++i)
  {
    i_prev = (i>0)?(i-1):(num-1);
    i_next = (i<num-1)?(i+1):0;
    // distance from each corner in C to the next corner so that edge_length(i) 
    // is the distance from C(i,:) to C(i+1,:) defined cyclically
    double edge_length = std::sqrt((C1.row(i) - C1.row(i_next)).dot(C1.row(i) - C1.row(i_next)));
    for (int j = 0; j<V1.rows(); ++j)
    {
      Eigen::VectorXd v = C1.row(i) - V1.row(j);
      Eigen::VectorXd vnext = C1.row(i_next) - V1.row(j);
      Eigen::VectorXd vprev = C1.row(i_prev) - V1.row(j);
      // distance from V to every C, where dist_C_V(i,j) is the distance from domain
      // vertex j to handle i
      dist_C_V(i,j) = std::sqrt(v.dot(v));
      double dist_C_V_next = std::sqrt(vnext.dot(vnext));
      double a_prev = std::atan2(vprev[1],vprev[0]) - std::atan2(v[1],v[0]);
      double a_next = std::atan2(v[1],v[0]) - std::atan2(vnext[1],vnext[0]);
      // mean value coordinates
      WW(i,j) = (std::tan(a_prev/2.0) + std::tan(a_next/2.0)) / dist_C_V(i,j);
      
      if (dist_C_V(i,j) < EPSILON)
        on_corner.push_back(std::make_pair(j,i));
      else
        // only in case of no-corner (no need for checking for multiple segments afterwards --
        // should only be on one segment (otherwise must be on a corner and we already
        // handled that)
        // domain vertex j is on the segment from i to i+1 if the distances from vj to
        // pi and pi+1 are about 
        if(abs((dist_C_V(i,j) + dist_C_V_next) / edge_length - 1) < EPSILON)
          on_segment.push_back(std::make_pair(j,i));
      
    }
  }
  
  // handle degenerate cases
  // snap vertices close to corners
  for (unsigned i = 0; i<on_corner.size(); ++i)
  {
    int vi = on_corner[i].first;
    int ci = on_corner[i].second;
    for (int ii = 0; ii<C.rows(); ++ii)
      WW(ii,vi) = (ii==ci)?1:0;
  }
  
  // snap vertices close to segments
  for (unsigned i = 0; i<on_segment.size(); ++i)
  {
    int vi = on_segment[i].first;
    int ci = on_segment[i].second;
    int ci_next = (ci<num-1)?(ci+1):0;
    for (int ii = 0; ii<C.rows(); ++ii)
      if (ii == ci)
        WW(ii,vi) = dist_C_V(ci_next,vi);
      else
      {
        if ( ii == ci_next)
          WW(ii,vi)  = dist_C_V(ci,vi);
        else
          WW(ii,vi) = 0;
      }
  }
  
  // normalize W
  for (int i = 0; i<V.rows(); ++i)
    WW.col(i) /= WW.col(i).sum();
  
  // we've made W transpose
  W = WW.transpose();
}

#endif