A Framework for Symmetric Self-Intersecting Surfaces

Abstract

3D printing of surfaces has become an established method for prototyping and visualisation. However, surfaces often contain certain degenerations, such as self-intersecting faces or non-manifold parts, which pose problems in obtaining a 3D printable file. Therefore, it is necessary to examine these degenerations beforehand. Surfaces in three-dimensional space can be represented as embedded simplicial complexes describing a triangulation of the surface. We use this combinatorial description, and the notion of embedded simplicial surfaces (which can be understood as well-behaved surfaces) to give a framework for obtaining 3D printable files. This provides a new perspective on self-intersecting triangulated surfaces in three-dimensional space. Our method first retriangulates a surface using a minimal number of triangles, then computes its outer hull, and finally treats non-manifold parts. To this end, we prove an initialisation criterion for the computation of the outer hull. We also show how symmetry properties can be used to simplify computations. Implementations of the proposed algorithms are given in the computer algebra system GAP4. To verify our methods, we use a dataset of self-intersecting symmetric icosahedra. Exploiting the symmetry of the underlying embedded complex leads to a notable speed-up and enhanced numerical robustness when computing a retriangulation, compared to methods that do not take advantage of symmetry.