I use the following relationships for converting Euler angles g={E1, E2, E3} to orientation matrix A=[a1 b1 c1; a2 b2 c2; a3 b3 c3] and vice versa:
a1=cos(E1)* cos(E3)- sin(E1)*sin(E3)*cos(E2);
b1=sin(E1)* cos(E3)+ cos(E1)*sin(E3)*cos(E2);
c1=sin(E3)* sin(E2);
a2=-cos(E1)* sin(E3)- sin(E1)*cos(E3)*cos(E2);
b2=-sin(E1)* sin(E3)+ cos(E1)*cos(E3)*cos(E2);
c2=cos(E3)* sin(E2);
a3=sin(E1)* sin(E2);
b3=-cos(E1)* sin(E2);
c3=cos(E2);
E1_calculated=atan(-A31/A32) ;
E2_calculated=acos(A33);
E3_calculated=atan(A13/A23);
I utilized E1, E2 and E3 from EBSD data and converted them to A matrix but in some cases, only in some cases, the calculated E1 is not correct. In fact, pi (3.14) should be added to it to produce the correct value of E1. For example:
Real values: E1(rad)= 2.575931; E2= 0.734958; E3= 0.479442;
Calculated values: E1=-0.5657; E2= 0.734958; E3= 0.479442;
Real values: E1=3.413515; E2=0.186401; E3= 0.955568;
Calculated values: E1=0.2719; E2=0.186401; E3= 0.955568;
What's the problem?
----- Matlab code of operations is attached----
Without analyzing the problem I can say that matrices lead to problems, Quaternions are better choice, but to be sure and have an elegant general formalism in all dimensions just use geometric algebra. It is powerful, there is no special cases and you will have a clear geometric picture at each step. The problem is in that you need double cover of rotation group, geometric algebra provides it. It is instructive to study the "gimbal lock"
Best wishes,
Miroslav
Dear Miroslav,
The Euler angles are not uniquely defined. Moreover, you may have discontinuities for some angles. It is better to use the rotation angle and the rotation axes by Rodrigues formula. See the discussion in the attached article.
Well, a simple application of trigonometric functions always cause some multiple solutions, however if correctly done you can use Euler angles as well...without any problems. However, exactly because of these conditions (symmetry of angles, e.g. 359+1=0 and not 360) an interpretation is not always easy and the averaging of different orientations in terms of angles like (359+1)/2=180 is simply wrong. This you always have to do with quaternions or RF-angles. Nevertheless, orientations are still described by Euler angles (surprizingly) although they have some serious disadvantages. I even think that no serious software is using Euler angles but transfers them immediately into quaternions for calculations. However, because of the even more complicated imagination Euler angles are still preferred compared to quaternions.
Article Euler angles and crystal symmetry
You need to use atan2 to compute the E1 angle instead of atan. See the documentation of atan2 in Matlab.
@Morteza ,
I'd like you to try geometric algebra, there is my paper (still unpublished) and you could find many interesting ideas and learn how to deal with rotations o'powerfully. But there is a lot of possibilities unit.
https://www.researchgate.net/publication/314671751_Multiplication_of_vectors_and_structure_of_3D_Euclidean_space?ev=prf_high
To rotate, reflect, use spinors etc just learn this natural system, you will be able work in any dimension, without matrices, tensors, quaternions and all that stuff. With matrices and other tables you have no any clear geometric meaning, but in GA you have, at each step. Imagine, for example, imaginary unit that have geometric meaning - defines the plane and direction.
Best wishes,
Miroslav
Article Multiplication of vectors and structure of 3D Euclidean space
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