Mitered offset of a mesh using QEM and vertex split

In this paper, we present a mitered offsetting method of a triangular mesh. Though our main target application is machining tool path generation, it can also be applied to shelling/hollowing of solid objects, collision avoidance in robot path planning, and so on. Previous literature on mesh offsetting mostly suggest inserting a portion of a cylinder (or a ball) in order to fill the gap between offset faces adjacent to a sharp edge (or a sharp vertex, respectively). The gap filling elements (cylinders or balls) are approximated by a number of small triangles depending on the offset error tolerance. Those small gap filling triangles not only increase tool path computation time, but also cause harmful effect in the accuracy of the machined result around the sharp edges. In this research, we try to reduce the number of gap filling triangles while meeting the given tolerance by introducing the concept of mitered offset, which is popularly used in 2D profile machining practice. We borrowed and modified the notion of quadric error metric (QEM) from the mesh simplification area. A modified version of QEM is used for robust computation of the offset vertex position which minimizes the sum of squared distance error from the faces around the original mesh vertex. If the error is within tolerance, the offset vertex is accepted. Otherwise, the offset vertex is split repeatedly until the error is acceptable. Vertex split occurs at the sharp features. A rigorous foundation is given to the mitered offset of 3D mesh with sharp features as well as smooth regions. The experimental results indicate that only a small number of triangles are added in offset mesh.