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Robust Laplace operators for general (possibly nonmanifold) triangle meshes and point clouds. For details, see A Laplacian for Nonmanifold Triangle Meshes by Nicholas Sharp and Keenan Crane from SGP 2020.

For Python bindings (with pip package), see

This method efficiently generates a high-quality V x V Laplace matrix for any (possibly nonmanifold, with or without boundary) triangular 3D surface mesh. In particular, the resulting Laplacian will always satisfy the maximum principle, with all-positive edge weights between nodes. The method can also produce a similar Laplacian for a point cloud.

The core algorithm is implemented in geometry-central. This is a simple application which loads a mesh or point cloud, builds our intrinsic tufted cover, and generates the resulting Laplace (and mass) matrix.

As a command-line tool, the program will output matrices in a simple format which can be read in by other programs (including MATLAB, etc). It can also be run with the --gui flag, to create a window and visualize the relevant data---though be wary that this visualization is significantly more expensive and less robust than the main algorithm.

GUI demo gif

Building and running

On unix-like machines, use:

git clone --recurse-submodules
cd nonmanifold-laplacian
mkdir build && cd build
cmake -DCMAKE_BUILD_TYPE=Release ..
make -j4
./bin/tufted-idt /path/to/your/mesh.obj

The codebase also builds on Visual Studio 2017 & 2019, by using CMake to generate a Visual Studio solution file.

The input should be a mesh or point cloud; any inputs with no faces will be processed as point clouds. Use the --gui flag to load a 3D gui to inspect the results.

If the input contains unreferenced vertices that do not appear in any face, the program will still ensure that the resulting matices have the same dimensions and indexing as the input vertex set by adding dummy rows. Optional parameters below control what values are inserted into these dummy rows; the defaults set 1 for the diagonal of the Laplacian and a small positive value for the diagonal of the mass matrix, to yield matrices which are full rank.


Use --help for defaults, etc.

flag purpose
--gui Show the GUI
--mollifyFactor Amount of intrinsic mollification to apply, relative to the mesh length scale. Larger values will lend robustness to floating-point degeneracy, though very large values will distort geometry. Reasonable range is roughly 0 to 1e-3. Default: 1e-6
--nNeigh Number of nearest-neighbors to be used for point cloud Laplacian. The construction is not very sensitive to this parameter, it usually does not need to be tweaked. Default: 30
--laplacianReplace For any unreferenced vertices in the input, put this this value in the diagonal of the Laplace matrix. Default: 1
--massReplace For any unreferenced vertices in the input, put this this value in the diagonal of the mass matrix. Negative values will interpreted relative to the smallest mass entry among referenced vertices, like X times the smallest mass. Default: -1e-3
--outputPrefix Prefix to prepend to all output file paths. Default: tufted_
--writeLaplacian Write the resulting Laplace matrix. A sparse VxV matrix, holding the weak Laplace matrix (that is, does not include mass matrix). Name: laplacian.spmat
--writeMass Write the resulting mass matrix. A sparse diagonal VxV matrix, holding lumped vertex areas. Name: lumped_mass.spmat

Output formats

Sparse matrices are output as an ASCII file where each line one entry in the matrix, giving the row, column, and value. The row and column indices are 1-indexed to make matlab happy. These files can be automatically loaded in matlab (see here). Parsers in other environments should be straightforward.

Known issues

This implementation is not the same code which was used to generate the results in the paper. If you need exact comparisons, please contact the authors.

Direct dependencies

(all are permissively licensed, and packaged with the repo)


  author={Nicholas Sharp and Keenan Crane},
  title={{A Laplacian for Nonmanifold Triangle Meshes}},
  journal={Computer Graphics Forum (SGP)},


A robust Laplace matrix for general (possibly nonmanifold) triangle meshes, and point clouds [Sharp & Crane SGP 2020]