Fast and robust mesh arrangements using floating-point arithmetic

Abstract

We introduce a novel algorithm to transform any generic set of triangles in 3D space into a well-formed simplicial complex. Intersecting elements in the input are correctly identified, subdivided, and connected to arrange a valid configuration, leading to a topologically sound partition of the space into piece-wise linear cells. Our approach does not require the exact coordinates of intersection points to calculate the resulting complex. We represent any intersection point as an unevaluated combination of input vertices. We then extend the recently introduced concept of indirect predicates [Attene 2020] to define all the necessary geometric tests that, by construction, are both exact and efficient since they fully exploit the floating-point hardware. This design makes our method robust and guaranteed correct, while being virtually as fast as non-robust floating-point based implementations. Compared with existing robust methods, our algorithm offers a number of advantages: it is much faster, has a better memory layout, scales well on extremely challenging models, and allows fully exploiting modern multi-core hardware with a parallel implementation. We thoroughly tested our method on thousands of meshes, concluding that it consistently outperforms prior art. We also demonstrate its usefulness in various applications, such as computing efficient mesh booleans, Minkowski sums, and volume meshes.

Automatic Surface Segmentation for Seamless Fabrication Using 4‐axis Milling Machines

“…Our approach has been implemented in C++, using cg3lib [MN21] and VCG [CNR13] for geometry processing functions, Eigen [GJ*14] for linear algebra routines, libigl [JP*16] and CGAL [FP09] for mesh Booleans, PyMeshLab [MC21] for mesh measurements. We plan in a future release to use the method described in [CLSA20] as soon as it is available in open source for Boolean operations since it promises to be more efficient. To ensure our approach's replicability, we will provide the source code of the application released with an open-source license.…”

Boundary-sampled halfspaces

“…However, our formulation and algorithms apply to other halfspace representations as well, such as closed meshes, parametric surfaces, and other types of implicit surfaces. To compute the arrangement of the halfspaces, we first tessellate the primitives and polygonize the implicit surfaces using Marching Cubes (we use a uniform grid of size 64 3 ), and we compute the mesh arrangement using the recent method of [Cherchi et al 2020].…”

“…Since we are computing a polygonal approximation of the shape's boundary, exact arrangements are not necessary, and we instead compute the arrangement after first polygonizing the boundary surface of each halfspace. This allows us to leverage recent development of robust and efficient algorithms for computing arrangements of closed meshes [Cherchi et al 2020;Zhou et al 2016].…”

“…Even if the halfspaces are planes, the time complexity of arrangement computation is cubic to the number of planes [Agarwal and Sharir 2000]. In our experiments, computing mesh arrangement using the latest mesh arrangement algorithm [Cherchi et al 2020] took over 5 Table 1. Comparing the number of samples generated by our algorithm for each 3D reverse engineering example with the number of connected components of the same halfspace (a lower bound) and the total number of patches on the shape boundary (an upper bound).…”

Convex polyhedral meshing for robust solid modeling

“…[Ju 2004]), whereas local approaches may be precise (e.g. [Cherchi et al 2020]) but their results may become invalid when coordinates are rounded to floating point values (see Sect 2.7). Local approaches may need to be complemented with hole filling [Zhao et al 2007] or mesh completion [Podolak and Rusinkiewicz 2005] algorithms to turn the valid simplicial complex to a closed polyhedron.…”