%% Rotations

% Explains how to define rotations and how to switch between different Euler

% angle conventions.

%

%% Open in Editor

%

%% Contents

%

%% Description

%

% Rotations are represented in MTEX by the class *rotation* which is 

% inherited from the class <quaternion_index.html 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

%

% * Bunge (phi1,Phi,phi2)       - ZXZ

% * Matthies (alpha,beta,gamma) - ZYZ

% * Roe (Psi,Theta,Phi)

% * Kocks (Psi,Theta,phi)

% * Canova (omega,Theta,phi)

%

% *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 two 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)





%%

% *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')





%% 

% *Changing the Default Euler Angle Convention*

%

% The default Euler angle convention can be changed by the command

% <set_mtex_option.html set_mtex_option>, for a permanent change the

% <matlab:edit('mtex_settings.m') mtex_settings> should be edited. Compare



set_mtex_option('EulerAngleConvention','Roe')

o



%%

set_mtex_option('EulerAngleConvention','Bunge')

o



%% Other Ways of Defining a Rotation

% 

% *The axis angle parametrisation*

%

% A very simple possibility to specify a rotation is to specify the 

% rotation axis and the rotation angle.



o = rotation('axis',xvector,'angle',30*degree)



%%

% *Four vectors defining a rotation*

%

% Given four vectors u1, v1, u2, v2 there is a unique rotation q such that

% q u1 = v1 and q u2 = v2.



o = rotation('map',xvector,yvector,zvector,zvector)



%%

% If only two vectors are specified, then the rotation with the smaller angle is

% returned and gives the rotation from first vector onto the second one.



o = rotation('map',xvector,yvector)



%%

% *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)





%%

% *Defining an rotation by a 3 times 3 matrix*



o = rotation('matrix',eye(3))



%%

% *Defining an 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)



%%

% Actually, MTEX represents internally every rotation as a quaternion.

% Hence, one can write



q = quaternion(1,0,0,0)

o = rotation(q)





%% Calculating with Rotations

%

% *Rotating Vectors*

%

% Let



o = rotation('Euler',90*degree,90*degree,0*degree)



%%

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



v = o * xvector



%%

% The inverse rotation is computed via the <rotation.mldivide.html backslash operator>



o \ v



%%

% *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



%%

% *Computing the rotation angle and the rotational axis*

%

% Then rotational angle and the axis

% of rotation can be computed via then commands

% <quaternion.angle.html angle(rot)> and

% <quaternion.axis.html axis(rot)>



angle(rot)/degree



axis(rot)



%%

% If two rotations are specifies the command 

% <quaternion.angle.html angle(rot1,rot2)> computes the rotational angle

% between both rotations



angle(rot1,rot2)/degree





%%

% *The inverse Rotation*

%

% The inverse rotation you get from the command

% <quaternion.inverse.html inverse(rot)>



inverse(rot)



%% Conversion into Euler Angles and Rodrigues Parametrisation

%

% There are methods to transform quaternion in almost any other

% parameterization of rotations as they are:

%

% * <quaternion.Euler.html, Euler(rot)>   in Euler angle

% * <quaternion.Rodrigues.html, Rodrigues(rot)>  in Rodrigues parameter

%



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





%% Plotting Rotations

%

% The <quaternion.plot.html plot> function allows you to visualize a

% rotation by plotting how the standard basis x,y,z transforms under the

% rotation.



cla reset;set(gcf,'position',[43   362   400   300])

plot(rot)