Here you can find a description of standard 3DEM conventions (as proposed by Heymann et al. JSB 151(2), 2005, p. 196-207 with corrections) and conventions used by popular software packages if different.

Coordinate convention is right-handed Cartesian coordinate system. Display convention is for x-axis to increase from left to right, y-axis to increase from bottom to top, and z-axis to increase from back to front (pointing at viewer) as shown:

A positive rotation is defined as clockwise for the object. A positive rotation is defined as anti-clockwise for the coordinate system when viewed with the axis of rotation pointing at the viewer. For example, a rotation from the x-axis to the y-axis (about the z-axis) is positive. Note that the object will rotate clockwise when the coordinate system rotates anti-clockwise.

Traditionally in EM the direction of propagation of the electron beam is thought to coincide with z axis, which in addition uniquely specifies xy as the plane in which the data is collected. Thus, it is convenient to express the rotations with respect to the z axis. By using a ZYZ convention (rotation around the z axis, followed by a rotation around y, and another around z), one benefits from the fact that the description bears a simple relation to the description of a point on a sphere; that is to say, we can think of the decomposition as the description of a point on the sphere (the first two Eulerian angles YZ) and a final in-plane rotation (the final Z-Eulerian angle). We denote three respective Euler angles as phi (φ), theta (θ), psi (ψ) and the corresponding rotation in matrix notation as a product of three matrices:

RZ(ψ) RY(θ) RZ(φ)

Euler angles are three successive axial rotations:

1. phi, a rotation about the z-axis;
2. theta, a rotation about the y'-axis; and
3. psi, a rotation about the z''-axis.

Rotations of coordinate system (anti-clockwise):

Recall that it is the matrix at the rightmost end that is applied first. So one might write a 3DEM rotation as (φ, θ, ψ), where φ is applied first, θ second and ψ lastly.

Based on definition of Euler angles above, it is easy to see that first two Eulerian angles (φ, θ) define projection direction, while ψ defines rotation of projection in-plane of projection. Thus, as far as projections are concerned ψ is a trivial angle, as its change does not change the 'information content' of the projection.

The range of possible Eulerian angles for an asymmetric structure is 0≤φ≤360, 0≤θ≤180, 0≤ψ≤360). However, for each projection whose direction is (φ, θ, ψ) there exists a projection that is related to it by an in-plane mirror operation along x-axis and whose direction is (180+φ, 180-θ, -ψ). Note the projection direction of the mirrored projection is also in the same range of Eulerian angles as all angles are given modulo 360 degrees (i.e., if say φ > 360, then φ = φ - 360, also if φ < 0, then φ = φ + 360.

Relion <-> FReAlign

• CTFFIND3 to IMAGIC

DEFOCUS1=DFMID1
DEFOCUS2=DFMID2
DEFANGLE=90-ANGAST

• CTFFIND3 to SPIDER (now CTFFIND3 is available from inside SPIDER)

defocus = (DFMID1 + DFMID2)/2;
astig = (DFMID2 - DFMID1);
angle_astig = ANGAST - 45;
if (astig < 0) {
    astig = -astig;
    angle_astig = angle_astig + 90;
}

• Relion to EMAN2 (use e2reliontoeman.py)

defocus=(rlnDefocusU+rlnDefocusV)/20000.0
dfang=rlnDefocusAngle
dfdiff=(rlnDefocusU-rlnDefocusV)/10000.0

• EMAN2 to Relion

  See e2refinetorelion2d and e2refinetorelion3d

• SPIDER to FReAlign

df1 = spider.defocus - spider.magastig/2
df2 = spider.defocus + spider.magastig/2
angast = spider.angast + 45

• EMAN1/2 boxer to XMIPP 2.4 Mark
• SPIDER Web to XMIPP 2.4

• e2proc2d / e2proc3d
• em2em
• dm3 to other formats
• Various IMOD commands
• relion_convert_to_tiff

• 3DEM conventions
• Few info about Euler angles
• The Transform Class in SPARX and EMAN2
• Transform Python class in EMAN2
• Convert Euler angles for projections
• https://github.com/alisterburt/eulerangles