Curvescope

A mean curvature viewer based on Polyscope.

Details

A simple C++ viewer for mean curvature vizualization of triangular meshes, built upon the 3D data manipulation and vizualization library Polyscope. Supports .obj, .ply, and .off files.

The implementation is based on "Discrete Differential-Geometry Operators for Triangulated 2-Manifolds, Meyer et al". Specifically, the mean curvature K(x_i) vector at each vertex of the mesh x_i is computed according to:

For the computation of the vertex area A_mixed two approaches are considered:

Voronoi vertex areas are computed according to the the algorithm described in Fig.4 of the afforementioned paper.
Barycentric areas are provided natively by Polyscope.

How-To

Compile & run

To download, run:

git clone --recurse-submodules https://github.com/alexdkeros/Curvscope.git

To compile the code, navigate to the project's directory and run:

mkdir build
cd build
cmake ..
make

The executable will be located in <project directory>/build/bin/curvscope.

How to use

In order to launch the viewer, from the project's directory, run:

./build/bin/curvscope [path/to/mesh1.obj path/to/mesh2.ply path/to/mesh3.off ...]

(Command line arguments are optional)

How to use the Command UI:

Dependencies

Curvscope only depends on Polyscope, which packages all its source code dependencies with the repository.

Polyscope is included as a git submodule.

Showcase

Standard examples

In the following examples mean curvature is computed with Mean Curvature (Voronoi areas):

Model	Vizualization	Comments
plane		Zero mean curvature at non-boundary vertices.
sphere		(Near-) constant curvature K=1.
torus		Smoothly decreasing curvature.
spot		Vizualization of mean curvature vectors. Unseen vectors may point inwards (for example, at the horns), or have zero magnitude.
happy buddha		"Big and complicated" mesh.

Voronoi vs barycentric vertex areas

The torus case

Mean Curvature (Voronoi areas)	Mean Curvature (Barycentric areas)

Barycentric areas can cause irregularities in curvature computation due to triangulation peculiarities, as seen below:

Voronoi vertex areas	Barycentric vertex areas

The sphere case

Similarly to the torus case, barycentric vertex areas on the sphere can cause outliers in curvature computation.

Mean Curvature (Voronoi areas)	Mean Curvature (Barycentric areas)

Mean curvature flow

A very crude implementation of mean curvature flow, according to the computed mean curvature vectors. Unfortunately, when face areas become too small singularities cause them to get deleted.

Before	After

Comparison with libIGL (alternative implementation)

An alternative approach would be to use libIGL, a handy geometry processing library. To this end, one only needs to clone the Polyscope-libIGL project template and add the following functions in main.cpp, with their respective callbacks. Examples of these functions are provided in the libIGL tutorial.

The Laplace-Beltrami operator approach:

void addMeanCurvatureScalar() {
  using namespace Eigen;
  using namespace std;

  MatrixXd HN;
  VectorXd H;
  SparseMatrix<double> L,M,Minv;


  igl::cotmatrix(meshV,meshF,L);
  igl::massmatrix(meshV,meshF,igl::MASSMATRIX_TYPE_VORONOI,M);
  igl::invert_diag(M,Minv);
  HN = -Minv*(L*meshV);
  H = HN.rowwise().norm(); //up to sign

  polyscope::getSurfaceMesh("input mesh")
      ->addVertexScalarQuantity("mean curvature", H,
                                polyscope::DataType::SYMMETRIC);

  polyscope::getSurfaceMesh("input mesh")
      ->addVertexVectorQuantity("mean curvature vectors", HN);

}

The Quadratic fit approach:

void addMeanCurvatureQuadFitScalar() {
  using namespace Eigen;
  using namespace std;

  VectorXd H;
  MatrixXd PD1,PD2;
  VectorXd PV1,PV2;

  igl::principal_curvature(meshV,meshF,PD1,PD2,PV1,PV2);
  // mean curvature
  H = 0.5*(PV1+PV2);

  polyscope::getSurfaceMesh("input mesh")
      ->addVertexScalarQuantity("mean curvature (QuadFit)", H,
                                polyscope::DataType::SYMMETRIC);


  polyscope::getSurfaceMesh("input mesh")
      ->addVertexVectorQuantity("Principal vec1", PD1);

  polyscope::getSurfaceMesh("input mesh")
      ->addVertexVectorQuantity("Principal vec2", PD2);

}

Timing comparisons on example meshes (average of 10 executions):

Mesh	Ours (Voronoi vertex areas)	Laplace-Beltrami operator	Quadratic fit
spot	3ms	118ms	2764ms
bunnyhead	1ms	67ms	1528ms

Example mean curvature vizualization on bunnyhead (colorscale range [0-20]):

Ours (Voronoi vertex areas)	Laplace-Beltrami operator	Quadratic fit