geometry/meshes/curvature-measures-nc-XY-3d.cpp

Computation of principal curvatures and directions on a torus mesh, using Normal cycle curvature measures (based on the theory of Normal cycle).

# a 20x20 discretized torus with a radius 0.5 for measures
./examples/geometry/meshes/curvature-measures-nc-XY-3d 20 20 0.5

outputs

Expected k1 curvatures: min=-0.5 max=0.25
Computed k1 curvatures: min=-0.581281 max=0.441977
Expected k2 curvatures: min=1 max=1
Computed k2 curvatures: min=0.904081 max=1.06404

It also produces several OBJ files to display curvature estimation results, example-nc-K1.obj, example-nc-D1.obj, example-nc-K2.obj, and example-nc-D2.obj as well as the associated MTL file.

Normal cycle smallest principal curvature and direction, r=0.5

Normal cycle greatest principal curvature and direction, r=0.5

See also

Curvature measures on meshes and digital surfaces

Note

Normal cycle theory fails on bad sampling of smooth surfaces. A well known example is the Schwarz lantern. You may try the following:

./examples/geometry/meshes/curvature-measures-nc-3d lantern 20 20 0.5

outputs

Expected k1 curvatures: min=0 max=0
Computed k1 curvatures: min=0.404148 max=0.417222
Expected k2 curvatures: min=0.5 max=0.5
Computed k2 curvatures: min=0.875976 max=1.85041

 
#include <iostream>
#include <algorithm>
#include "DGtal/base/Common.h"
#include "DGtal/math/linalg/EigenDecomposition.h"
#include "DGtal/shapes/SurfaceMesh.h"
#include "DGtal/shapes/SurfaceMeshHelper.h"
#include "DGtal/geometry/meshes/NormalCycleComputer.h"
#include "DGtal/io/writers/SurfaceMeshWriter.h"
#include "DGtal/io/colormaps/GradientColorMap.h"
#include "DGtal/io/colormaps/QuantifiedColorMap.h"
#include "DGtal/helpers/Shortcuts.h"
 
DGtal::GradientColorMap< double >
makeColorMap( double min_value, double max_value )
{
  DGtal::GradientColorMap< double > gradcmap( min_value, max_value );
  gradcmap.addColor( DGtal::Color( 0, 0, 255 ) );
  gradcmap.addColor( DGtal::Color( 0, 255, 255 ) );
  gradcmap.addColor( DGtal::Color( 255, 255, 255 ) );
  gradcmap.addColor( DGtal::Color( 255, 255, 0 ) );
  gradcmap.addColor( DGtal::Color( 255, 0, 0 ) );
  return gradcmap;
}
 
void usage( int argc, char* argv[] )
{
  std::cout << "Usage: " << std::endl
            << "\t" << argv[ 0 ] << " <shape> <m> <n> <R>" << std::endl
            << std::endl
            << "Computation of principal curvatures and directions on a shape, " << std::endl
            << "using Normal Cycle anisotropic curvature measure." << std::endl
            << "- builds a <shape> in {torus,lantern,sphere}, with     " << std::endl
            << "  <m> latitude points and <n> longitude points."         << std::endl
            << "- <R> is the radius of the measuring balls."             << std::endl
            << "It produces several OBJ files to display principal "     << std::endl
            << "curvatures and directions estimations: `example-cnc-K1.obj`" << std::endl
            << "`example-cnc-K2.obj`, `example-cnc-D1.obj`, and"         << std::endl
            << "`example-cnc-D2.obj` as well as associated MTL files."   << std::endl;
}
 
int main( int argc, char* argv[] )
{
  if ( argc <= 1 )
    {
      usage( argc, argv );
      return 0;
    }
  using namespace DGtal;
  using namespace DGtal::Z3i;
  typedef SurfaceMesh< RealPoint, RealVector >          SM;
  typedef NormalCycleComputer< RealPoint, RealVector >  NC;
  typedef SurfaceMeshHelper< RealPoint, RealVector >    SMH;
  // a shape in "torus|lantern|sphere"
  std::string input = argv[ 1 ];
  int    m = argc > 2 ? atoi( argv[ 2 ] ) : 20;  // nb latitude points
  int    n = argc > 3 ? atoi( argv[ 3 ] ) : 20;  // nb longitude points
  double R = argc > 4 ? atof( argv[ 4 ] ) : 0.5; // radius of measuring ball
 
  SM smesh;
  double exp_K1_min = 0.0;
  double exp_K1_max = 0.0;
  double exp_K2_min = 0.0;
  double exp_K2_max = 0.0;
  if ( input == "torus" )
    {
      const double big_radius   = 3.0;
      const double small_radius = 1.00001; // avoid codacy warnings
      smesh = SMH::makeTorus( big_radius, small_radius,
                              RealPoint { 0.0, 0.0, 0.0 }, m, n, 0,
                              SMH::NormalsType::VERTEX_NORMALS );
      exp_K1_min = ( 1.0 / ( small_radius - big_radius ) );
      exp_K1_max = ( 1.0 / ( big_radius + small_radius ) );
      exp_K2_min = 1.0 / small_radius;
      exp_K2_max = 1.0 / small_radius;
    }
  else if ( input == "sphere" )
    {
      const double radius = 2.0;
      smesh = SMH::makeSphere( radius, RealPoint { 0.0, 0.0, 0.0 }, m, n,
                               SMH::NormalsType::VERTEX_NORMALS );
      exp_K1_min = 1.0 / radius;
      exp_K1_max = 1.0 / radius;
      exp_K2_min = 1.0 / radius;
      exp_K2_max = 1.0 / radius;
    }
  else if ( input == "lantern" )
    {
      const double radius = 2.0;
      smesh = SMH::makeLantern( radius, 1.0, RealPoint { 0.0, 0.0, 0.0 }, m, n,
                                SMH::NormalsType::VERTEX_NORMALS );
      exp_K1_min = 0.0;
      exp_K1_max = 0.0;
      exp_K2_min = 1.0 / radius;
      exp_K2_max = 1.0 / radius;
    }
 
  // builds a NormalCycleComputer object onto the torus mesh
  NC nc( smesh );
  // computes area, anisotropic XY curvature measures
  auto mu0  = nc.computeMu0();
  auto muXY = nc.computeMuXY();
 
  // Estimates principal curvatures (K1,K2) and directions (D1,D2) by
  // measure normalization and eigen decomposition.
  std::vector< double > K1( smesh.nbFaces() );
  std::vector< double > K2( smesh.nbFaces() );
  std::vector< RealVector > D1( smesh.nbFaces() );
  std::vector< RealVector > D2( smesh.nbFaces() );
  // Principal directions computation requires a local face normal
  smesh.computeFaceNormalsFromPositions();
  for ( auto f = 0; f < smesh.nbFaces(); ++f )
    {
      const auto b    = smesh.faceCentroid( f );
      const auto N    = smesh.faceNormals()[ f ];
      const auto area = mu0 .measure( b, R, f );
      const auto M    = muXY.measure( b, R, f );
      std::tie( K1[ f ], K2[ f ], D1[ f ], D2[ f ] )
        = nc.principalCurvatures( area, M, N );
    }
 
  auto K1_min_max = std::minmax_element( K1.cbegin(), K1.cend() );
  auto K2_min_max = std::minmax_element( K2.cbegin(), K2.cend() );
  std::cout << "Expected k1 curvatures:"
            << " min=" << exp_K1_min << " max=" << exp_K1_max
            << std::endl;
  std::cout << "Computed k1 curvatures:"
            << " min=" << *K1_min_max.first << " max=" << *K1_min_max.second
            << std::endl;
  std::cout << "Expected k2 curvatures:"
            << " min=" << exp_K2_min << " max=" << exp_K2_max
            << std::endl;
  std::cout << "Computed k2 curvatures:"
            << " min=" << *K2_min_max.first << " max=" << *K2_min_max.second
            << std::endl;
 
  typedef SurfaceMeshWriter< RealPoint, RealVector > SMW;
  typedef Shortcuts< KSpace > SH;
  const auto colormapK1 = makeQuantifiedColorMap( makeColorMap( -0.625, 0.625 ) );
  const auto colormapK2 = makeQuantifiedColorMap( makeColorMap( -0.625, 0.625 ) );
  auto colorsK1 = SMW::Colors( smesh.nbFaces() );
  auto colorsK2 = SMW::Colors( smesh.nbFaces() );
  for ( auto i = 0; i < smesh.nbFaces(); i++ )
    {
      colorsK1[ i ] = colormapK1( K1[ i ] );
      colorsK2[ i ] = colormapK2( K2[ i ] );
    }
  SMW::writeOBJ( "example-nc-K1", smesh, colorsK1 );
  SMW::writeOBJ( "example-nc-K2", smesh, colorsK2 );
  const auto avg_e = smesh.averageEdgeLength();
  SH::RealPoints positions( smesh.nbFaces() );
  for ( auto f = 0; f < positions.size(); ++f )
    {
      D1[ f ] *= smesh.localWindow( f );
      positions[ f ] = smesh.faceCentroid( f ) - 0.5 * D1[ f ];
    }
  SH::saveVectorFieldOBJ( positions, D1, 0.05 * avg_e, SH::Colors(),
                          "example-nc-D1",
                          SH::Color::Black, SH::Color( 0, 128, 0 ) );
  for ( auto f = 0; f < positions.size(); ++f )
    {
      D2[ f ] *= smesh.localWindow( f );
      positions[ f ] = smesh.faceCentroid( f ) - 0.5 * D2[ f ];
    }
  SH::saveVectorFieldOBJ( positions, D2, 0.05 * avg_e, SH::Colors(),
                          "example-nc-D2",
                          SH::Color::Black, SH::Color(128, 0,128 ) );
  
  return 0;
}