erot

Rotation matrix from Euler angles.

Syntax

Rp = erot(angles)
Rp = erot(alpha,beta,gamma)

[xc,yc,zc] = erot(...,'cols')
[xr,yr,zr] = erot(...,'rows')

Description

erot returns an 3x3 rotation matrix in Rp. The 3-element vector angles=[alpha beta gamma] defines the three Euler angles for the rotation in radians. They can be also specified as three separate arguments alpha, beta and gamma.

For a detailed description of the associated rotation, the convention used, and mathematical formulas see the page on relative orientations.

Basically, erot provides a transformation matrix for vectors and matrices. If the Euler angles describe the transformation of the coordinate frame 1 to the coordinate frame 2, then you can use Rp to transform a vector v or tensor A from coordinates 1 to coordinates 2.

  v2 = Rp*v1;        % transform v from frame 1 to frame 2 representation
  A2 = Rp*A1*Rp.';   % transform A from frame 1 to frame 2 representation

This does not rotate the vector or matrix, but transforms them from one coordinate system representation to the other. After the transformation, the vector/matrix is still the same, only its representation has changed. Often, such a coordinate transformation is called a passive rotation.

Ra = Rp.' is the corresponding active transformation. The active rotation is also a composition of three rotations: first by -gamma around z, then by -beta around y, and last by -alpha around z. In this case it is the vector or matrix rather than the coordinate system which changes orientation.

If the argument 'cols' is given, erot returns the three columns of the rotation matrix separately. The three vectors and the rotation matrix are related by Rp = [xc yc zc]. Similarly, if the argument 'rows' is given, erot returns the three rows of the rotation matrix separately. However, they are returned not as rows but as column vectors. These three vectors and the rotation matrix are related by Rp = [xr.'; yr.'; zr.'].

The inverse of erot, i.e. calculating the Euler angles from a given rotation matrix, is provided by eulang.

Examples

To rotate the basis of the matrix A so that the final Z axis is along -z, and (X,Y) = (-y,-x), type

A = [1 2 3; 4 5 6; 7 8 9];
Rp = erot(pi/2,pi,0);
A1= Rp*A*Rp.'

A1 =
    5.0000    4.0000    6.0000
    2.0000    1.0000    3.0000
    8.0000    7.0000    9.0000

To rotate a magnetic field vector from the z direction to a direction in the first quadrant specified by the polar angles theta (down from z axis, elevation complement) and phi (counterclockwise around z from x, azimuth), use an active rotation.

B = [0;0;350];
theta = 20;
phi = 75;
Rp = erot([phi theta 0]*pi/180);
Ra = Rp.';
B_rotated = Ra*B

B_rotated =
   30.9825
  115.6281
  328.8924

To get the three columns separately, use

[xc,yc,zc] = erot([10 20 40]*pi/180,'cols')

You can see that these are the three columns of the rotation matrix

R = erot([10 20 40]*pi/180)

R =
    0.5973    0.7580   -0.2620
   -0.7279    0.6495    0.2198
    0.3368    0.0594    0.9397

Algorithm

See relative orientations and coordinate framesfor details.