One crucial stage was the introduction of specific mathematical operations, such as reflections, rotations, and translations, that are used to describe with precision how the parts are to be exchanged.
If the symmetry group contains reflection and rotation or more than one rotation, all possible combinations need to be included in the combined dataset.
For the analysis of symmetry under reflection and rotation of order 6, the symmetry group includes twelve symmetry transformations: each of the six rotations of the preceding analysis is now included with or without reflection.
The PC1 represents the first type of shape change, which is symmetric under reflection and rotation of order 2. The PC2 stands for the second group of shape changes, which is symmetric under rotation of order 2 but not under reflection.
"Definitely!" responded Joshua Zucker, "A lot of people see and comment on the reflection or rotation symmetry of the 5×5 grid, but not so many people point out that you could just as well be saving 5 tame wolves from 3 vicious attack sheep".
The Procrustes approach matches one configuration to another and produces a measure of match, while seeking the isotropic dilation and the rigid translation, reflection and rotation needed to best match one configuration to another (Cox and Cox, 1994).
The block-diagonal property of the nodal response matrix is determined by the symmetry properties of the hexagonal node in geometry, material and basis functions, including both reflection and rotation symmetries.
Under any reflection or rotation ρ, the delta function is invariant: :\delta(\rho \mathbf{x}) = \delta(\mathbf{x}).
Procedures were designed to yield features invariant to translation, scale, reflection, and rotation in the xy-plane.
