Euler angles: Background
A general rotation of a rigid body (e.g. a 3D reconstruction) can be described as a series of 3 rotations about the axes of the coordinate system (i.e. x, y and z axes). The angles of the 3 rotations are known as Euler angles (a subset are also known as Tait-Bryan angles).
Each rotation can be around one of three axes. Given that successive axes must be different, there are 3 x 2 x 2 = 12 possibilities. Although in practice only a few axis combinations are used, this can still be a source of confusion. In addition, a positive angle may be interpreted as clockwise or anti-clockwise around the axis (when viewed in a specified direction).
Rotations can be described in terms of external fixed axes, or internal axes of the rigid body which rotate with the body. It can be shown that a set of intrinsic rotations about axes x then y then z through angles α, β, γ is equivalent to a set of extrinsic rotations about axes z then y then x by angles γ, β, α.
A common convention is to rotate α about the z-axis, β about the new y-axis, and γ about the final z-axis. These are rotations with respect to an internal coordinate system. As above, this is equivalent to γ about external z, β about external y, and α about external z. In terms of rotation matrices, this would be written: Rot = R(Z,α)R(Y,β)R(Z,γ) for the case of rotation matrices pre-multiplying column vector coordinates.
General references:
Conventions used in cryoEM
Definitions of nomenclature:
| XYZ convention | Rotate about X then Y then Z with respect to an internal coordinate system. The conversion to an external coordinate system is described above. |
| α,β,γ | Successive angles of rotation about an internal coordinate system. |
| Clockwise | Positive angle is clockwise rotation when viewed outwards along positive axis (clearly it will be anti-clockwise if viewed inwards). |
| Beta | Should be in the range 0 to pi. This can be achieved using the equivalence (α,β,γ) = (π+α,-β,π+γ). |
Known conventions
| PEET | ZXZ |
| SPIDER | ZYZ anti-clockwise. (α,β,γ) is (ψ,θ,φ), note reversal because Spider defines its angles w.r.t external axes. |
| SPARX | Claims to follow Spider convention: ZYZ and (α,β,γ) is (ψ,θ,φ). However, the documentation seems to suggest a clockwise convention for angles. |
| BSOFT | ZYZ |
| XMIPP | ZYZ clockwise. (α,β,γ) is (φ,θ,ψ) |
| FREALIGN | ZYZ. (α,β,γ) is (rot,tilt,ψ) |
| RELION | ZYZ. Conventions are the same as in XMIPP and FREALIGN |
| IMAGIC | ZYZ anti-clockwise |
| EMAN2 | ZYZ |