Rotations (The Class rotation)

This section describes the class rotation and gives an overview on how to work with rotations in MTEX.

On this page ...
Class Description
Euler Angle Conventions
Other Ways of Defining a Rotation
Calculating with Rotations
Conversion into Euler Angles and Rodrigues Parametrisation
Plotting Rotations
Complete Function list

Class Description

The class rotation is an inheritance of the class quaternion and allow to work with rotations as with matrixes in MTEX.

Euler Angle Conventions

There are several ways to specify a rotation in MTEX. A well-known possibility are the so called Euler angles. In texture analysis the following conventions are commonly used

Defining a Rotation by Bunge Euler Angles

The default Euler angle convention in MTEX are the Bunge Euler angles. Here a rotation is determined by three consecutive rotations, the first about the z-axis, the second about the y-axis, and the third again about the z-axis. Hence, one needs three angles to define an rotation by Euler angles. The following command defines a rotation by its three Bunge Euler angles

o = rotation('Euler',30*degree,50*degree,10*degree)
 
o = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
    30   50   10    0
 
 

Defining a Rotation by Other Euler Angle Conventions

In order to define a rotation by a Euler angle convention different to the default Euler angle convention you to specify the convention as an additional parameter, e.g.

o = rotation('Euler',30*degree,50*degree,10*degree,'Roe')
 
o = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
   120   50  280    0
 
 

Changing the Default Euler Angle Convention

The default euler angle convention can be changed by the command setpref, for a permanent change the mtex_settings should be edited. Compare

setMTEXpref('EulerAngleConvention','Roe')
o
 
o = rotation  
  size: 1 x 1
 
  Roe Euler angles in degree
  Psi Theta   Phi  Inv.
   30    50    10     0
 
 
setMTEXpref('EulerAngleConvention','Bunge')
o
 
o = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
   120   50  280    0
 
 

Other Ways of Defining a Rotation

The axis angle parametrisation

A very simple possibility to specify a rotation is to specify the rotational axis and the rotational angle.

o = rotation('axis',xvector,'angle',30*degree)
 
o = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
     0   30    0    0
 
 

Four vectors defining a rotation

Given four vectors u1, v1, u2, v2 there is a unique rotations q such that q u1 = v1 and q u2 = v2.

o = rotation('map',xvector,yvector,zvector,zvector)
 
o = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
    90    0    0    0
 
 

If only two vectors are specified the rotation with the smallest angle is returned that maps the first vector onto the second one.

o = rotation('map',xvector,yvector)
 
o = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
    90    0    0    0
 
 

A fibre of rotations

The set of all rotations that rotate a certain vector u onto a certain vector v define a fibre in the rotation space. A discretisation of such a fibre is defined by

u = xvector;
v = yvector;
o = rotation('fibre',u,v)
 
o = rotation  
  size: 1 x 361
 

Defining a rotation by a 3 times 3 matrix

o = rotation('matrix',eye(3))
 
o = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
     0    0    0    0
 
 

Defining a rotation by a quaternion

A last possibility is to define a rotation by a quaternion, i.e., by its components a,b,c,d.

o = rotation('quaternion',1,0,0,0)
 
o = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
     0    0    0    0
 
 

Actually, MTEX represents internally every rotation as a quaternion. Hence, one can write

q = quaternion(1,0,0,0)
o = rotation(q)
 
q = Quaternion  
  size: 1 x 1
  a b c d
  1 0 0 0
 
o = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
     0    0    0    0
 
 

Calculating with Rotations

Rotating Vectors

Let

o = rotation('Euler',90*degree,90*degree,0*degree)
 
o = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
    90   90    0    0
 
 

a certain rotation. Then the rotation of the xvector is computed via

v = o * xvector
 
v = vector3d  
 size: 1 x 1
  x y z
  0 1 0

The inverse rotation is computed via the backslash operator

o \ v
 
ans = vector3d  
 size: 1 x 1
  x y z
  1 0 0

Concatenating Rotations

Let

rot1 = rotation('Euler',90*degree,0,0);
rot2 = rotation('Euler',0,60*degree,0);

be two rotations. Then the rotation defined by applying first rotation one and then rotation two is computed by

rot = rot2 * rot1
 
rot = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
     0   60   90    0
 
 

Computing the rotation angle and the rotational axis

Then rotational angle and the axis of rotation can be computed via then commands angle(rot) and axis(rot)

angle(rot)/degree

axis(rot)
ans =
  104.4775
 
ans = vector3d  
 size: 1 x 1
         x         y         z
  0.447214 -0.447214  0.774597

If two rotations are specified the command angle(rot1,rot2) computes the rotational angle between both rotations

angle(rot1,rot2)/degree
ans =
  104.4775

The inverse Rotation

The inverse rotation you get from the command inv(rot)

inv(rot)
 
ans = rotation  
  size: 1 x 1
 
  Bunge Euler angles in degree
  phi1  Phi phi2 Inv.
    90   60  180    0
 
 

Conversion into Euler Angles and Rodrigues Parametrisation

There are methods to transform quaternion in almost any other parameterization of rotations as they are:

[alpha,beta,gamma] = Euler(rot,'Matthies')
alpha =
    4.7124
beta =
    1.0472
gamma =
    3.1416

Plotting Rotations

The plot function allows you to visualize an rotation by plotting how the standard basis x,y,z transforms under the rotation.

plot(rot)

Complete Function list

charquaternion to char
displaystandart output
dotcompute rot1 . rot2
dot_outer
linedraw rotations connected by lines
matrixquaternion to direction cosine matrix conversion
mldivideo \ v
mtimesr = a * b
permuteoverloads permute
powerr.^n
project2FundamentalRegionprojects rotation to a fundamental region
repmatoverloads repmat
reshapeoverloads reshape
rotationdefines an rotation
subSetindexing of rotation
subsasgnoverloads subsasgn
subsrefoverloads subsref
timesr = a .* b
transposearray of rotations
uminusimplements rotation
uniquedisjoint list of rotations