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.

翻译：三维打印曲面已成为原型制作和可视化的一种成熟方法。然而，曲面通常包含某些退化特征，例如自相交面或非流形部分，这些特征在获取可三维打印文件时会造成问题。因此，有必要预先检查这些退化特征。三维空间中的曲面可表示为描述曲面三角剖分的嵌入单纯复形。我们利用这种组合描述以及嵌入单纯曲面（可理解为行为良好的曲面）的概念，建立了一个获取三维可打印文件的框架。这为理解三维空间中自相交三角剖分曲面提供了新的视角。我们的方法首先使用最少数量的三角形对曲面进行重新三角剖分，然后计算其外凸壳，最后处理非流形部分。为此，我们证明了外凸壳计算的初始化准则。我们还展示了如何利用对称性来简化计算。所提出算法的实现在计算机代数系统GAP4中给出。为验证我们的方法，我们使用了自相交对称二十面体的数据集。与未利用对称性的方法相比，在计算重新三角剖分时，利用底层嵌入复形的对称性可显著提高计算速度并增强数值鲁棒性。