Collaborative Computational Project
for Electron cryo-Microscopy

MRC

Rotation conventions

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:

Wikipedia

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