Z axis rotation by successive X axis and Y axis rotations?

[tb12]

Rotation about the Z-axis by successive rotations about the X-axis and the Y-axis?

My original question was poorly/imprecise phrased. I'll try to clarify/rephrase:

Arbitrary rotation about the Z-axis using only successive rotations about the X-axis and the Y-axis?

Consider an object (with some arbitrary shape) in a standard 3-dimensional X-Y-Z coordinate system (right-handed, cartesian).

If you first rotate the object 90 degrees about the X-axis, then rotate the object 90 degrees about the Y-axis and then rotate the object -90 degrees about the X-axis, the result will be the same as if you had only rotated the object 90 degrees about the Z-axis.

Can it be generalised to any angle of rotation about the Z-axis?

E.g. how to perform a 45 degree rotation about the Z-axis without actually rotating about the Z-axis but instead using only any number of succesive rotations about the X-axis and the Y-axis?

The following is my original post. Please forget it and read above.


Z axis rotation by successive X axis and Y axis rotations?
For example:
First 90 degree X axis rotation
then 90 degree Y axis rotation
then -90 degree X axis rotation
results in 90 degree rotation about Z axis.
How to get 45 degree Z axis rotation by only using successive rotations around X axis and Y axis?
Can this be generalised to any Z angle?

 

[Ishuda]

For example:
First 90 degree X axis rotation
then 90 degree Y axis rotation
then -90 degree X axis rotation
results in 90 degree rotation about Z axis.

How to get 45 degree Z axis rotation by only using successive rotations around X axis and Y axis?

Can this be generalised to any Z angle?

First 90 degree X axis rotation - what are you calling a 90 degree rotation, the x axis into the y (or -y) axis or into z (-z) axis.

Similarly with the other rotations.

 

[tb12]

First 90 degree X axis rotation - what are you calling a 90 degree rotation, the x axis into the y (or -y) axis or into z (-z) axis.

Similarly with the other rotations.

I have rephrased my question in the first post. I hope it more understandable now.