The origin of objects is always related to whatever symmetry it has:
An object with any form of symmetry is oriented such that a symmetry axis is parallel to the z-axis.
The origin is defined as the point where several symmetry axes intersect, expressed as the offset from the first pixel or voxel in the image.
A symmetry axis lies on the z-axis in all cases, in most it is the major symmetry axis.
The origin is defined as the point in the image grid where symmetry axes intersect and has pixel or voxel units.
Schoenflies notation is used to identify point groups.
Table 4.1. Point group symmetry conventions
| Symmetry | Notation | Origin | Orientation |
|---|---|---|---|
| Asymmetric | C1 | User-defined | user-defined |
| Cyclic | C<n> | On symmetry axis | n-fold axis on z-axis |
| Dihedral | D<n> | Intersection of symmetry axes | n-fold axis on z-axis, 2-fold axis on x-axis |
| Tetrahedral | T | Intersection of symmetry axes | 2-fold axes on x, y, and z-axes |
| Octahedral/Cubic | O | Intersection of symmetry axes | 4-fold axes on x, y, and z-axes |
| Icosahedral/Dodecahedral | I | Intersection of symmetry axes | 2-fold axes on x, y, and z-axes, front 5-fold axes in yz plane |
where <n> is the symmetry order of the major axis
of the cyclic and dihedral point groups.
Icosahedral symmetry have two commonly used orientations: The front most 5-fold axes may lie in the yz plane (consistent with X-ray crystallographic convention), or they may lie in the xz plane (consistent with some EM packages, notably PFT and EM3DR). The first is the preferable orientation, indicated by the symbol I, while the second is 90 degrees rotated from the first and indicated by I90.
Helical symmetry is indicated by up to four parts:
Rise per asymmetric unit (translation along the helical axis oriented on the z-axis)
Rotation angle around the helical axis per asymmetric unit
Presence of a dyad axis (perpendicular to the helical axis and oriented on the x-axis)
Cyclic symmetry around the helical axis
The notation is: H<rise>,<angle>,<dyad>,<n>