# A way to represent a rotation matrix as a rotation around an x, y and z-axis without Euler angles

In our clinical research, we want to show the difference in the pose of two superimposed forearm bones. To be interpretable for a surgeon, we want to show these rotational differences as rotations around 3 axes. However, Euler angles are not suitable because multiple solutions exist. So we want to describe the rotation when only looking from the front, only from the side, and only from above.

A solution may be to use an axis angel representation with a normalized axis. To show the rotation around the x-axis, the x component of the axis is multiplied by the 3D rotation. By doing so, the rotation is weighted for the x component. This method can by no means describe the transformation between the bones but does describe the isolated differences around each axis.

This seems to work fairly well, but I cannot find any reference to this concept. The only place I could find it is a calculator where it is mentioned as: "Axis with angle magnitude " https://www.andre-gaschler.com/rotationconverter/.   Is there any literature that used this to describe a rotation (not too complex)? What is this called? Or is there a better way to do this?

To give some more context: We are comparing the pose of a left wrist bone to a right wrist bone. We are interested in the differences in those bones, because differences in bones, results in differences in function. flexion of the wist is likely correlated with a rotation around the y axis of the bone. Therefore, we want to know component of the rotation solely around the Y-axis. (the same goes for the X and Z axis)

kind regards

• Daniel90

I don't think the offered bounty is high enough to motivate users to spend time on this question.

• Mathe

I agree with Daniel

• Tonekim

I increased the bounty.

• Nirenberg

So the rotation about x,y,z-axis are given, and you want to do rotations about x,y, z-axis and need the rotation matrix?

• Tonekim

No. A rotation matrix is given, and I want to know the 'isolated' X,Y and Z rotations. So the question is: How much of the 3d rotation takes place around the Xaxis? (And y and z axis.) So the rotations should not add up to do the full transformation.

• Nirenberg

It is then a fairly involved problem, and would take over an hour to answer.

• Tonekim

Then I won't have the budget for it. I hoped any of you would have known the concept of "Axis with angle magnitude" as I described and could help me on some literature,

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