Abstract—Marching Cubes is a popular choice for isosurface extraction from regular grids due to its simplicity, robustness, and efficiency. One of the key shortcomings of this approach is the quality of the resulting meshes, which tend to have many poorly shaped and degenerate triangles. This issue is often addressed through postprocessing operations such as smoothing. As we demonstrate in experiments with several data sets, while these improve the mesh, they do not remove all degeneracies and incur an increased and unbounded error between the resulting mesh and the original isosurface. Rather than modifying the resulting mesh, we propose a method to modify the grid on which Marching Cubes operates. This modification greatly increases the quality of the extracted mesh. In our experiments, our method did not create a single degenerate triangle, unlike any other method we experimented with. Our method incurs minimal computational overhead, requiring at most twice the execution time of the original Marching Cubes algorithm in our experiments. Most importantly, it can be readily integrated in existing Marching Cubes implementations and is orthogonal to many Marching Cubes enhancements (particularly, performance enhancements such as out-of-core and acceleration structures). Index Terms—Meshing, marching cubes. Ç 1

Most computational codes that use irregular grids depend on the triangle quality of the single worst triangle in the grid: skinny triangles can lead to bad performance and numerical instabilities. Marching Cubes is the standard isosurface grid generation algorithm, and while most triangles it generates are good, it almost always generates some bad triangles. Here we show how simple changes to Marching Cubes can lead to a drastically reduced number of degenerate triangles, making it a more practical choice for isosurface grid generation, reducing or eliminating the need and costs of post-processing. 1.

Three-dimensional meshes are frequently used to perform physical simulations in science and engineering. This involves decomposing a domain into a mesh of small elements, usually tetrahedra or hexahedra. The elements must be of good quality; in particular there should be no plane or dihedral angle close to 0 or 180 degrees. Automatically creating such meshes for complicated domains is a challenging problem, especially guaranteeing good dihedral angles, a goal that has eluded researchers for nearly two decades. By using point lattices, notably the body centered cubic lattice, we develop two tetrahedral mesh generation algorithms that, for the first time, come with meaningful guarantees on the quality of the elements. For domains bounded by an isosurface, we generate a tetrahedral mesh whose dihedral angles are bounded between 10.7 and 164.8 degrees, or (with a change in parameters) between 8.9 and 158.8 degrees. The algorithm is numerically robust and easy to implement because it generates tetrahedra from a small set of precomputed stencils. The algorithm is so fast that it can be invoked at each time step of a simulation, possibly in real time for small meshes. The tetrahedra are uniformly sized on the boundary, but in the interior it is