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A Framework for Symmetric Self-Intersecting Surfaces

Christian Amend, Tom Goertzen

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.

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.
Paper Structure (11 sections, 6 theorems, 35 equations, 17 figures, 1 table, 4 algorithms)

This paper contains 11 sections, 6 theorems, 35 equations, 17 figures, 1 table, 4 algorithms.

Table of Contents

  1. Introduction
  2. Embedded Simplicial Surfaces
  3. Detecting Self-Intersections
  4. Retriangulation of Self-Intersecting Complexes
  5. Symmetric Optimisation for Retriangulation
  6. Computation of Outer Hull and Chambers
  7. Non-Manifold Parts
  8. Experiments on Self-Intersecting Icosahedra
  9. Conclusion and Outlook
  10. Acknowledgements
  11. Appendix

Key Result

Proposition 3.1

Let $f$ be a triangle with vertices $v_1, v_2, v_3 \in \mathbb{R}^3$ and normal vector $n_f$, and consider the planes A point $p$ lies inside $f$ if and only if $p$ lies on the same plane as $f$ and satisfies the following condition: where $\tilde{n}_i$ is a normal vector of the plane $P_i$, pointing towards the interior of $f$.

Figures (17)

  • Figure 1: (a) Identifying two cubes at an edge leads to a non-manifold edge. In 3D printing applications, this edge is often neglected, leading to two separated cubes. (b) Two intersecting cubes are shown together with a view of its interior. The interior part can be omitted by reduction to the outer-hull.
  • Figure 2: (a) We can modify the two cubes in Figure \ref{['fig:TouchingCubes']} by operations replacing the part containing the non-manifold edge. When 3D printing the adapted surface, the two cubes are connected. (b) For 3D printing applications, only the outer-hull is printed. Thus, we obtain a modified surface by neglecting inner parts.
  • Figure 3: (a) The great icosahedron has 12 embedded vertices, 30 edges and 20 faces, and the same incidence structure as the regular icosahedron (a Platonic solid). (b,c) Each of the 20 faces intersects with 15 faces non-trivially, and we can compute the intersections of all face pairs by only considering one face (face in red) and in a next step use the symmetry group to compute the remaining intersections.
  • Figure 4: (a) Intersections of one face of the great icosahedron with other faces up to symmetry. (b) The stellation diagram of a face of the great icosahedron, showing lines where other face planes intersect with this one coxeter1982fiftynine, can be obtained from (a) by rotations and reflections. (c) Computing all intersection points of intersection within the face. (d) Triangulating non-triangle parts, yielding a simplicial disc. This retriangulation can be carried over to all 20 faces of the great icosahedron using its symmetry group in order to obtain a surface without self-intersections.
  • Figure 5: Exploded views of the $413$ internal chambers of the great icosahedron with different magnitudes $m$.
  • ...and 12 more figures

Theorems & Definitions (40)

  • Definition 2.1: Simplicial Complex
  • Definition 2.2
  • Definition 2.3: Simplicial surface
  • Definition 2.4: Embedding
  • Remark 2.5
  • Definition 2.6: Orientation
  • Remark 2.7
  • Definition 2.8: Chambers and outer hull
  • Definition 2.9
  • Definition 2.10: Intersection points
  • ...and 30 more