6.3.2 Handling Small Angles

Figure 6.11: Issues with small angles

For sets with a high rotational symmetry order, small angles occur near the rotation center. The smallest angle of the triangle colored in red is $22.5\degree$.

Extracting slices of rotationally symmetric objects with a high symmetry order $n$ might lead to small angles near the rotation center as visualized in Figure 6.11. In that case, a regular $n$-polygon $P \subseteq A$ with its center being the rotation center can be extruded from $A$ to form a separate block. To minimize newly introduced angles, $P$ should be rotationally aligned in a way that the slice starting angle $\sigma$ of $A$ passes through a vertex of $P$. The remaining set $A \setminus P$ is still rotationally symmetric with order $n$, but the small angle in the center is eliminated. Using this approach, the geometry can be represented by a templated geometry having two templates, being the polygon $P$ around the rotation center and a slice of $A \setminus P$.

Figure 6.12: Small angle optimization for rotational symmetries

A regular eight-polygon is decomposed in two different ways. The smallest inner angle of template $X_1$ (top picture) is $45\degree$, while the smallest inner angle of template $X_2$ and $X_3$ is $67.5\degree$ which is significantly larger and therefore enables better element qualities (bottom picture).

Figure 6.13: Small angle optimization for high rotational symmetry orders $n$.

This process is visualized in Figure 6.12. The structure instance of a templated mesh, which is generated based on that templated geometry, is potentially not rotationally symmetric around its center any more. However, no angle of $360\degree/n$ is introduced at the center of the slice but two angles equal to $(180\degree+\alpha)/2 = (n+2)/(2n) 180\degree$ are introduced (cf. Figure 6.13). These angle are larger than the slice angle of $360\degree/n$ which would have been introduced in the center. This potentially leads to a better quality of the mesh elements in the slice instances. The mesh generation of the center template is not required to be rotationally symmetric and the mesh generation algorithm therefore has more freedom when creating volumetric elements hence leading to better overall mesh element quality. The approach also works for 3D sets and geometries. However, the set $P$, which is cut out around the axis, is not a regular $n$-prism in general.