One-Shot Method for Computing Generalized Winding Numbers

C++ implementation of One-Shot Method for Computing Generalized Winding Numbers (Martens & Bessmeltsev, CGF 2025).

This repository hosts a reference implementation of both the 3D and 2D code.

Citation

@article{Martens2025WindingNumberOneShot,
  title   = {One-Shot Method for Computing Generalized Winding Numbers},
  author  = {Martens, Cedric and Bessmeltsev, Mikhail},
  journal = {Computer Graphics Forum},
  doi     = {10.1111/cgf.70194},
  volume  = {44},
  number  = {5},
  year    = {2025},
}

2D code

Informal Blog Post

Build

Needs a C++17 compiler, CMake >= 3.20, and Eigen 3.

cmake -S . -B build -DCMAKE_BUILD_TYPE=Release
cmake --build build --target wn2d wn2d_tests wn2d_benchmark winding_field_plot

Run the tests and the examples:

ctest --test-dir build -R wn2d --output-on-failure
./build/2d/examples/wn2d_benchmark
./build/2d/examples/winding_field_plot   # writes a PPM image

Using it from CMake

add_subdirectory(path/to/one_shot_wn/2d)
target_link_libraries(my_app PRIVATE wn2d)

Or pull it in with FetchContent and SOURCE_SUBDIR 2d.

Example

#include <wn2d/winding_number.h>

const double k = 0.5522847498;   // cubic-Bézier circle constant
std::vector<wn2d::BezierCurve> circle = {
    {{{ {1,0}, {1,k}, {k,1}, {0,1} }}},
    {{{ {0,1}, {-k,1}, {-1,k}, {-1,0} }}},
    {{{ {-1,0}, {-1,-k}, {-k,-1}, {0,-1} }}},
    {{{ {0,-1}, {k,-1}, {1,-k}, {1,0} }}}
};

double w_in  = wn2d::winding_number(circle, {0.0, 0.0});
double w_out = wn2d::winding_number(circle, {2.0, 0.0});

Eigen::MatrixXd W = wn2d::winding_number_grid(
    circle, -1.5, 1.5, -1.5, 1.5, /*resolution*/ 256);

The full API is in 2d/include/wn2d/winding_number.h.

3D code

Build

3D dependencies: libigl (submodule), GSL, CGAL (pulled via libigl), Boost (for Gauss-Kronrod quadrature), and OpenMP (recommended).

git submodule update --init --recursive
cmake -S . -B build -DCMAKE_BUILD_TYPE=Release
cmake --build build

This produces a static library gwn3d, the CLI one_shot_wn_3d, and the example/test binaries.

Running the examples and tests

./build/3d/examples/mesh_demo
./build/3d/examples/parametric_demo
./build/3d/examples/minimal_surface_demo
./build/3d/examples/coons_demo
./build/3d/examples/bezier_patch_demo
./build/3d/examples/point_query_demo
./build/3d/examples/compare_with_libigl
./build/3d/tests/gwn3d_tests

Linking from CMake

add_subdirectory(path/to/one_shot_wn/3d)
target_link_libraries(my_app PRIVATE gwn3d)