text: // This file is part of libigl, a simple c++ geometry processing library.
  //
  // Copyright (C) 2017 Amir Vaxman <avaxman@gmail.com>
  //
  // This Source Code Form is subject to the terms of the Mozilla Public License
  // v. 2.0. If a copy of the MPL was not distributed with this file, You can
  // obtain one at http://mozilla.org/MPL/2.0/.
  
  #include <igl/shapeup.h>
  #include <igl/min_quad_with_fixed.h>
  #include <igl/igl_inline.h>
  #include <igl/setdiff.h>
  #include <igl/cat.h>
  #include <igl/PI.h>
  #include <Eigen/Core>
  #include <vector>
  
  namespace igl
  {
      
    //This projection does nothing but render points into projP. Mostly used for "echoing" the global step
    IGL_INLINE bool shapeup_identity_projection(const Eigen::PlainObjectBase<Eigen::MatrixXd>& P, const Eigen::PlainObjectBase<Eigen::VectorXi>& SC, const Eigen::PlainObjectBase<Eigen::MatrixXi>& S,  Eigen::PlainObjectBase<Eigen::MatrixXd>& projP){
      projP.conservativeResize(SC.rows(), 3*SC.maxCoeff());
      for (int i=0;i<S.rows();i++){
        Eigen::RowVector3d avgCurrP=Eigen::RowVector3d::Zero();
        for (int j=0;j<SC(i);j++)
          avgCurrP+=P.row(S(i,j))/(double)(SC(i));
    
        for (int j=0;j<SC(i);j++)
          projP.block(i,3*j,1,3)=P.row(S(i,j))-avgCurrP;
      }
      return true;
    }
    
    
    //the projection assumes that the sets are vertices of polygons in order
    IGL_INLINE bool shapeup_regular_face_projection(const Eigen::PlainObjectBase<Eigen::MatrixXd>& P, const Eigen::PlainObjectBase<Eigen::VectorXi>& SC, const Eigen::PlainObjectBase<Eigen::MatrixXi>& S,  Eigen::PlainObjectBase<Eigen::MatrixXd>& projP){
      projP.conservativeResize(SC.rows(), 3*SC.maxCoeff());
      for (int currRow=0;currRow<SC.rows();currRow++){
      //computing average
        int N=SC(currRow);
        const Eigen::RowVectorXi SRow=S.row(currRow);
        Eigen::RowVector3d avgCurrP=Eigen::RowVector3d::Zero();
        Eigen::MatrixXd targetPolygon(N, 3);
        Eigen::MatrixXd sourcePolygon(N, 3);
        for (int j=0;j<N;j++)
          avgCurrP+=P.row(SRow(j))/(double)(N);
    
        for (int j=0;j<N;j++)
          targetPolygon.row(j)=P.row(SRow(j))-avgCurrP;
    
        //creating perfectly regular source polygon
        for (int j=0;j<N;j++)
          sourcePolygon.row(j)<<cos(2*igl::PI*(double)j/(double(N))), sin(2*igl::PI*(double)j/(double(N))),0.0;
    
        //finding closest similarity transformation between source and target
        Eigen::MatrixXd corrMat=sourcePolygon.transpose()*targetPolygon;
        Eigen::JacobiSVD<Eigen::Matrix3d> svd(corrMat, Eigen::ComputeFullU | Eigen::ComputeFullV);
        Eigen::MatrixXd R=svd.matrixU()*svd.matrixV().transpose();
        //getting scale by edge length change average. TODO: by singular values
        Eigen::VectorXd sourceEdgeLengths(N);
        Eigen::VectorXd targetEdgeLengths(N);
        for (int j=0;j<N;j++){
          sourceEdgeLengths(j)=(sourcePolygon.row((j+1)%N)-sourcePolygon.row(j)).norm();
          targetEdgeLengths(j)=(targetPolygon.row((j+1)%N)-targetPolygon.row(j)).norm();
        }
        double scale=(targetEdgeLengths.cwiseQuotient(sourceEdgeLengths)).mean();
    
        for (int j=0;j<N;j++)
          projP.block(currRow,3*j,1,3)=sourcePolygon.row(j)*R*scale;
      }
    
      return true;
    }
  
    template <
    typename DerivedP,
    typename DerivedSC,
    typename DerivedS,
    typename Derivedw>
    IGL_INLINE bool shapeup_precomputation(const Eigen::PlainObjectBase<DerivedP>& P,
                                           const Eigen::PlainObjectBase<DerivedSC>& SC,
                                           const Eigen::PlainObjectBase<DerivedS>& S,
                                           const Eigen::PlainObjectBase<DerivedS>& E,
                                           const Eigen::PlainObjectBase<DerivedSC>& b,
                                           const Eigen::PlainObjectBase<Derivedw>& wShape,
                                           const Eigen::PlainObjectBase<Derivedw>& wSmooth,
                                           ShapeupData & sudata)
    {
        using namespace std;
        using namespace Eigen;
        sudata.P=P;
        sudata.SC=SC;
        sudata.S=S;
        sudata.b=b;
        typedef typename DerivedP::Scalar Scalar;
        
        //checking for consistency of the input
        assert(SC.rows()==S.rows());
        assert(SC.rows()==wShape.rows());
        assert(E.rows()==wSmooth.rows());
        assert(b.rows()!=0);  //would lead to matrix becoming SPD
        
        sudata.DShape.conservativeResize(SC.sum(), P.rows());  //Shape matrix (integration);
        sudata.DClose.conservativeResize(b.rows(), P.rows());  //Closeness matrix for positional constraints
        sudata.DSmooth.conservativeResize(E.rows(), P.rows());  //smoothness matrix
          
        //Building shape matrix
        std::vector<Triplet<Scalar> > DShapeTriplets;
        int currRow=0;
        for (int i=0;i<S.rows();i++){
            Scalar avgCoeff=1.0/(Scalar)SC(i);
              
            for (int j=0;j<SC(i);j++){
              for (int k=0;k<SC(i);k++){
                if (j==k)
                  DShapeTriplets.push_back(Triplet<Scalar>(currRow+j, S(i,k), (1.0-avgCoeff)));
                else
                  DShapeTriplets.push_back(Triplet<Scalar>(currRow+j, S(i,k), (-avgCoeff)));
              }
            }
          currRow+=SC(i);
          
        }
   
        sudata.DShape.setFromTriplets(DShapeTriplets.begin(), DShapeTriplets.end());
  
        //Building closeness matrix
        std::vector<Triplet<Scalar> > DCloseTriplets;
        for (int i=0;i<b.size();i++)
          DCloseTriplets.push_back(Triplet<Scalar>(i,b(i), 1.0));
        
        sudata.DClose.setFromTriplets(DCloseTriplets.begin(), DCloseTriplets.end());
        
        //Building smoothness matrix
        std::vector<Triplet<Scalar> > DSmoothTriplets;
        for (int i=0; i<E.rows(); i++) {
          DSmoothTriplets.push_back(Triplet<Scalar>(i, E(i, 0), -1));
          DSmoothTriplets.push_back(Triplet<Scalar>(i, E(i, 1), 1));
        }
          
        SparseMatrix<Scalar> tempMat;
        igl::cat(1, sudata.DShape, sudata.DClose, tempMat);
        igl::cat(1, tempMat, sudata.DSmooth, sudata.A);
          
        //weight matrix
        vector<Triplet<Scalar> > WTriplets;
          
        //one weight per set in S.
        currRow=0;
        for (int i=0;i<SC.rows();i++){
            for (int j=0;j<SC(i);j++)
                WTriplets.push_back(Triplet<double>(currRow+j,currRow+j,sudata.shapeCoeff*wShape(i)));
            currRow+=SC(i);
        }
          
        for (int i=0;i<b.size();i++)
            WTriplets.push_back(Triplet<double>(SC.sum()+i, SC.sum()+i, sudata.closeCoeff));
          
        for (int i=0;i<E.rows();i++)
            WTriplets.push_back(Triplet<double>(SC.sum()+b.size()+i, SC.sum()+b.size()+i, sudata.smoothCoeff*wSmooth(i)));
          
        sudata.W.conservativeResize(SC.sum()+b.size()+E.rows(), SC.sum()+b.size()+E.rows());
        sudata.W.setFromTriplets(WTriplets.begin(), WTriplets.end());
          
        sudata.At=sudata.A.transpose();  //for efficieny, as we use the transpose a lot in the iteration
        sudata.Q=sudata.At*sudata.W*sudata.A;
  
        return min_quad_with_fixed_precompute(sudata.Q,VectorXi(),SparseMatrix<double>(),true,sudata.solver_data);
    }
  
  
    template <
    typename DerivedP,
    typename DerivedSC,
    typename DerivedS>
    IGL_INLINE bool shapeup_solve(const Eigen::PlainObjectBase<DerivedP>& bc,
                                  const std::function<bool(const Eigen::PlainObjectBase<DerivedP>&, const Eigen::PlainObjectBase<DerivedSC>&, const Eigen::PlainObjectBase<DerivedS>&,  Eigen::PlainObjectBase<DerivedP>&)>& local_projection,
                                  const Eigen::PlainObjectBase<DerivedP>& P0,
                                  const ShapeupData & sudata,
                                  const bool quietIterations,
                                  Eigen::PlainObjectBase<DerivedP>& P)
    {
      using namespace Eigen;
      using namespace std;
      MatrixXd currP=P0;
      MatrixXd prevP=P0;
      MatrixXd projP;
      
      assert(bc.rows()==sudata.b.rows());
      
  		MatrixXd rhs(sudata.A.rows(), 3); rhs.setZero();
      rhs.block(sudata.DShape.rows(), 0, sudata.b.rows(),3)=bc;  //this stays constant throughout the iterations
          
      if (!quietIterations){
          cout<<"Shapeup Iterations, "<<sudata.DShape.rows()<<" constraints, solution size "<<P0.size()<<endl;
          cout<<"**********************************************************************************************"<<endl;
      }
      projP.conservativeResize(sudata.SC.rows(), 3*sudata.SC.maxCoeff());
      for (int iter=0;iter<sudata.maxIterations;iter++){
        
        local_projection(currP, sudata.SC,sudata.S,projP);
              
        //constructing the projection part of the (DShape rows of the) right hand side
        int currRow=0;
        for (int i=0;i<sudata.S.rows();i++)
          for (int j=0;j<sudata.SC(i);j++)
            rhs.row(currRow++)=projP.block(i, 3*j, 1,3);
        
        DerivedP lsrhs=-sudata.At*sudata.W*rhs;
        MatrixXd Y(0,3), Beq(0,3);  //We do not use the min_quad_solver fixed variables mechanism; they are treated with the closeness energy of ShapeUp.
        min_quad_with_fixed_solve(sudata.solver_data, lsrhs,Y,Beq,currP);
        
        double currChange=(currP-prevP).lpNorm<Infinity>();
        if (!quietIterations)
          cout << "Iteration "<<iter<<", integration Linf error: "<<currChange<< endl;
        prevP=currP;
        if (currChange<sudata.pTolerance){
          P=currP;
          return true;
        }
      }
      
      P=currP;
      return false;  //we went over maxIterations
      
    }
  }
  
  
  
  
  
  #ifdef IGL_STATIC_LIBRARY
  template bool igl::shapeup_precomputation< typename Eigen::Matrix<double, -1, -1, 0, -1, -1>, typename Eigen::Matrix<int, -1, 1, 0, -1, 1>, typename Eigen::Matrix<int, -1, -1, 0, -1, -1>, typename Eigen::Matrix<double, -1, 1, 0, -1, 1> >(Eigen::PlainObjectBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> > const&, Eigen::PlainObjectBase<Eigen::Matrix<int, -1, 1, 0, -1, 1> > const&, Eigen::PlainObjectBase<Eigen::Matrix<int, -1, -1, 0, -1, -1> > const&, Eigen::PlainObjectBase<Eigen::Matrix<int, -1, -1, 0, -1, -1> > const&, Eigen::PlainObjectBase<Eigen::Matrix<int, -1, 1, 0, -1, 1> > const&, Eigen::PlainObjectBase<Eigen::Matrix<double, -1, 1, 0, -1, 1> > const&,   Eigen::PlainObjectBase<Eigen::Matrix<double, -1, 1, 0, -1, 1> > const&, igl::ShapeupData&);
  
  template bool igl::shapeup_solve<typename Eigen::Matrix<double, -1, -1, 0, -1, -1>, typename Eigen::Matrix<int, -1, 1, 0, -1, 1>, typename Eigen::Matrix<int, -1, -1, 0, -1, -1> >(const Eigen::PlainObjectBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> >& bc, const std::function<bool(const Eigen::PlainObjectBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> >&, const Eigen::PlainObjectBase<Eigen::Matrix<int, -1, 1, 0, -1, 1> >&, const Eigen::PlainObjectBase<Eigen::Matrix<int, -1, -1, 0, -1, -1> >&,  Eigen::PlainObjectBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> >& ) >& local_projection, const Eigen::PlainObjectBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> >& P0, const igl::ShapeupData & sudata, const bool quietIterations, Eigen::PlainObjectBase<Eigen::Matrix<double, -1, -1, 0, -1, -1> >& P);
  #endif
  