Conversion of rotation axes

Questions
1

Hi
I want to rotate my coordinate system in 45 degrees around Z axis, and after that 45 degree around Y axis.
My question is:
How I can do that, to be on the same orientation, if I can rotate only the X-axis and Y-axis?

Thanks.

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2

You can not do it by just rotating around X and Y (in any order). You would have to rotate also around Z:

In[1]:= mZY = RotationMatrix[45. Degree, {0, 0, 1}] . RotationMatrix[45. Degree, {0, 1, 0}]
Out[1]= {{0.5, -0.707107, 0.5}, {0.5, 0.707107, 0.5}, {-0.707107, 0., 0.707107}}

In[2]:= EulerAngles[mZY, {1, 2, 3}]/Degree
Out[2]= {-35.2644, 30., 54.7356}

In[3]:= EulerAngles[mZY, {2, 1, 3}]/Degree
Out[3]= {35.2644, -30., 35.2644}
3

Thanks for the answer.

Where can I find the Euler function?

4

You can find it, and several other related functions, in the math software Mathematica.

After all, you posted your question in a Mathematica forum!

5

Actually, one can get the same orientation if rotations are restricted to be around X and Y axes. But three rotations are needed.

In[1]:= (* ZY coordinate frame *)
mZY = EulerMatrix[{45. Degree, 45. Degree, 0}, {3, 2, 1}]
Out[1]= {{0.5, -0.707107, 0.5}, {0.5, 0.707107, 0.5}, {-0.707107, 0., 0.707107}}

In[2]:= (* XYX angles *)
EulerAngles[mZY, {1, 2, 1}]/Degree
Out[2]= {35.2644, 60., -54.7356}

In[3]:= (* YXY angles *)
EulerAngles[mZY, {2, 1, 2}]/Degree
Out[3]= {-90., 45., 135.}

The YXY angles can also be 90, -45, -45.

6

Thank you very much! Very helpful!

Can you be more specific - where I can find this calculator?

7

I use the software Wolfram Mathematica installed on a PC. It is a product that you pay for, but well worth the price. Discounts for students and other non-professional use. Here is info.

You can try it out on the free Wolfram Cloud