Level Sets of Weak-Morse Functions for Triangular Mesh Slicing

Abstract

In the context of CAD CAM CAE (Computer-Aided Design, Manufacturing and Engineering) and Additive Manufacturing, the computation of level sets of closed 2-manifold triangular meshes (mesh slicing) is relevant for the generation of 3D printing patterns. Current slicing methods rely on the assumption that the function used to compute the level sets satisfies strong Morse conditions, rendering incorrect results when such a function is not a Morse one. To overcome this limitation, this manuscript presents an algorithm for the computation of mesh level sets under the presence of non-Morse degeneracies. To accomplish this, our method defines weak-Morse conditions, and presents a characterization of the possible types of degeneracies. This classification relies on the position of vertices, edges and faces in the neighborhood outside of the slicing plane. Finally, our algorithm produces oriented 1-manifold contours. Each contour orientation defines whether it belongs to a hole or to an external border. This definition is central for Additive Manufacturing purposes. We set up tests encompassing all known non-Morse degeneracies. Our algorithm successfully processes every generated case. Ongoing work addresses (a) a theoretical proof of completeness for our algorithm, (b) implementation of interval trees to improve the algorithm efficiency and, (c) integration into an Additive Manufacturing framework for industry applications.

Vector Arithmetic in the Triangular Grid

“…Although we work with continuous coordinate system which uses real numbers, our work is essential in discrete mathematics, especially, in digital geometry to work, e.g., with digital images on the triangular grid. We should also note that hexagonal, triangular, honeycomb and other related grid structures are used in various other fields, e.g., in networks [ 12 , 13 , 17 , 18 , 19 ], in fractional calculus [ 20 , 21 ], in 3D printing [ 22 ], in chemical and physical modelling [ 23 ] and simulations [ 24 , 25 ], and in city planning [ 26 ], where continuous transformations play also crucial roles, thus our result may be applied. Additionally to the above mentioned fields, triangular grid is applied in skeletonization and thinning algorithms [ 27 , 28 ], in discrete tomography [ 29 ] and in cartography.…”