coordinate-based mesh manipulation/deformation using libigl

Question

I am currently using libigl to manipulate/deform a simple (2D) square by manipulating its corners (cf. initial state below):

After imposing displacement u(1,1) to the bottom-left corner and updating non-corner vertices locations using biharmonic deformation I end up with the following state (inspired from Biharmonic deformation tutorial ):

Problems are:

• triangles are not uniformly distributed
• The deformation has altered the sides which are no longer straight lines

An ideal solution would then be something close to the image below, without having to manage each boundary vertice:

Questions are:

• Which king of deformation method should I use to achieve such result?
• Is libigl the right tool for such operations?

NOTE: full implementation is:

#include <iostream>
#include <Eigen/Core>
#include <igl/colon.h>
#include <igl/harmonic.h>
#include <igl/viewer/Viewer.h>
#include <igl/triangle/triangulate.h>
#include "tutorial_shared_path.h"

using namespace Eigen;
using namespace std;

double step = 0.1;
// Input polygon
MatrixXd V; // corners location
MatrixXi E; // boundary edges
MatrixXd H; // 2D positions of points contained in holes of the triangulation

// Triangulated interior
MatrixXd D_bc; // displacements / corner
MatrixXd V2; // all vertexes location
MatrixXd U; // deformed vertexes
MatrixXi F2;
Eigen::VectorXi b; // index of corner vertices
int selected = 0; // index of currently selected corner

void update(igl::viewer::Viewer &viewer) {
  cout << "D_bc " <<endl
       << D_bc <<endl;
  // compute displacements
  MatrixXd D;
  igl::harmonic(V2,F2,b,D_bc,2.,D);
  // compute solution as init state + displacements
  U = V2+D;
  viewer.data.set_vertices(U);

  // display current selection
  Eigen::MatrixXd points(1,3);
  points << V(selected, 0) + D_bc(selected, 0), V(selected, 1) + D_bc(selected, 1), 0;
  viewer.data.set_points(points,Eigen::RowVector3d(1,0,0));
}

bool key_down(igl::viewer::Viewer &viewer, unsigned char key, int mods) {
  switch((int)key)
  {
    case 32: // space
      selected = (selected + 1) % 4;
      break;
    case 6: // right
      D_bc(selected, 0) += step;
      break;
    case 7: // left
      D_bc(selected, 0) -= step;
      break;
    case 8: // down
      D_bc(selected, 1) -= step;
      break;
    case 9: // top
      D_bc(selected, 1) += step;
      break;
    default:
      return false;
  }
  update(viewer);
  return true;
}

int main(int argc, char *argv[])
{
  // Create the boundary of a square
  V.resize(4,2);
  V << -1,-1, 1,-1, 1,1, -1,1;
  E.resize(4,2);
  E << 0,1, 1,2, 2,3, 3,0;
  H.resize(0,2);
  // Triangulate the interior
  igl::triangle::triangulate(V,E,H,"a0.01q",V2,F2);

  igl::colon<int>(0,V.rows()-1,b);

  // Plot the generated mesh
  igl::viewer::Viewer viewer;
  viewer.data.set_mesh(V2,F2);

  // set initial displacements / solution
  D_bc = MatrixXd::Zero(V.rows(), V.cols());
  U=V2;

  viewer.callback_key_down = &key_down;
  update(viewer);

  // display legend
  cout<<
    "Press [space] to switch point."<<endl<<
    "Use arrows to move current point."<<endl;

  viewer.launch();
}