# Euler angles in SPIDER & Web

The Euler angles in SPIDER & Web are defined as three
successive rotations in a right hand coordinate system.

- First, the object is rotated CLOCKWISE around the Z-axis
(angle 'phi')
- Then it is rotated CLOCKWISE around the original
Y-axis (angle 'theta')
- Finally, it is rotated CLOCKWISE around the original
Z-axis (angle 'psi').

All rotations are done around axes of the original SPACE coordinate
system and direction of rotation is determined by looking to the origin.

Note: If a volume is displayed in Web as slices, the observed rotations
will be, COUNTERCLOCKWISE for 'phi' and 'psi' rotations around Z-axis and
CLOCKWISE for 'theta' rotation around Y-axis. Web displays a volume
with first slice on top.

The rotation matrices used are defined as:
v = Rv', where R is the matrix for transforming vector v' to vector v.
R = R(psi) * R(theta) * R(phi)
R(psi) = cos(psi) sin(psi) 0
-sin(psi) cos(psi) 0
0 0 1
R(theta) = cos(theta) 0 -sin(theta)
0 1 0
sin(theta) 0 cos(theta)
R(phi) = cos(phi) sin(phi) 0
-sin(phi) cos(phi) 0
0 0 1

### Rotation by: Phi: 40 Theta: 50 & Psi: 70

### Animation Notes:

An object consisting of 3 cylinders of different lengths and
diameters that coincide with the three orthogonal axes was
created using SPIDER.

The RT 3D SPIDER operation was
used to rotate this object by Phi: 40 Theta: 50 & Psi: 70. A
cylinder parallel to the rotation axis is embedded to the object
before each rotation. Each rotation is done in 10 incremental steps
to capture the position of the object for making the
illustration. X', Y' and Z' are in the body coordinate system whereas Y
and Z are in the space coordinate system.

Source file: euler.html
Updated: 23 Mar. 2015

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