3.3.1 Symmetry Detection

Detection of symmetries is easy for linear geometries in ${\mathbb{R}}^2$. The initially devised algorithms create string literals based on the points and lines of the linear geometries for elements with the same distance to the center of gravity and using circular rotations and reversal of these literals to detect rotational and reflective symmetries [94]. These approaches can be extended to linear 3D geometries, however, additional information, like the axis of symmetry, is required [58]. Another approach uses surface sampling and clustering to detect symmetries for 2D and 3D geometries [98]. This algorithm is also capable of detecting not only exact but also approximate symmetries. A technique called generalized moment has also proven to be robust for identifying symmetries of a geometry in ${\mathbb{R}}^3$ [26]. This approach uses a spherical harmonics representation to efficiently find extrema of the generalized moment, which indicate potential symmetry candidates. However, a post processing step has to validate these candidates to filter out false positives. A similar algorithm uses the planar-reflective symmetry transform for detecting reflective symmetries [63]. The latter three algorithms can also be used for approximate symmetry detection.