About the nodal line in defining Euler angles

[ftft]

I hope someone can explain this to me. In the definition of Euler angles, the body-fixed-azimuthal angel γ is measured from the nodal line that defines the intersection of the body-fixed-XY plane and the space fixed-xy plane to the body-fixed-Y axis. This is the green line in this image from google

and I am using here the same labels as in the image. Now, if the body-fixed-XYZ axes are rotated in space, say, about the space-fixed-z axis, How should one define the new nodal line? In other words, how one should measure the body-fixed-azimuthal angel γ after the rotation?

 

[stevendaryl]

I'm not sure I understand your question, but let me try to make the geometric definition of the Euler angles a little more concrete.

Let's start with a globe, and single out three points on the globe:

1. The North Pole, which is latitude 90 (north).
2. The "West Pole", which is the point on the equator at latitude 0, longitude 0
3. The "East Pole", point on the equator at latitude 0, longitude 90 (east)

(nobody calls these points the East Pole or the West Pole, but I'm just using these names to have handy ways to refer to them.)
Let the [itex]X[/itex] axis be the line running from the equator through the West Pole. Let the [itex]Y[/itex] axis be the line running from the equator through the East Pole. Let the [itex]Z[/itex] axis be the line running from the equator through the North Pole. These lines are body-fixed. For some reason, the X-axis is also called the "line of nodes".

Pick three perpendicular (space-fixed) lines that intersect in one point and call them the x-axis, the y-axis and the z-axis. To start with, position the globe so that [itex]X[/itex] axis coincides with the x-axis, the [itex]Y[/itex] axis coincides with the y-axis and the [itex]Z[/itex] axis coincides with the z-axis.

Step 1: Rotate the globe about the Z axis through an angle [itex]\alpha[/itex]. (Positive [itex]\alpha[/itex] means rotating the East Pole toward the West Pole, and negative means rotating away from the West Pole. ([itex]\alpha[/itex] is between +90 and -90, in degrees)
Step 2: Rotate the globe about the (new location of the) X axis through an angle [itex]\beta[/itex]. Note that during this rotation, the line of nodes, which is the X axis, stays fixed. [itex]\beta[/itex] is always positive--it's the angle that the North Pole moves through as it is rotated down toward the x axis. [itex]\beta[/itex] is between 0 and 180 degrees.
Step 3: Rotate the globe again about the (new location of the) Z axis through an angle [itex]\gamma[/itex]. The angle [itex]\gamma[/itex] can be either positive or negative, like [itex]\alpha[/itex]. It's positive if you are rotating the West Pole toward the East Pole, and negative if you are rotating in the opposite direction. Again, [itex]\gamma[/itex] is between 0 and 180.

Now, your question was:

Now, if the body-fixed-XYZ axes are rotated in space, say, about the space-fixed-z axis, How should one define the new nodal line?

The nodal line is the same as the body-fixed X-axis. The idea of the Euler angles is that no matter what the orientation of the X-Y-Z axes are, you can always get that orientation by starting with the X-Y-Z axes lined up with the x-y-z axes and performing the three steps above (for the appropriate choice of [itex]\alpha, \beta, \gamma[/itex]).

 

[ftft]

Thanks a lot for the reply. I do understand the definition of Euler angles. I feel like the way I put my question was not very clear so let me rephrase it:

what happens when we rotate the body itself with its attached XYZ axes about the space-fixed-z axis? For example in the above figure, let's rotate XYZ about space-fixed-z axis by π/2. This can be viewed as rotating the blue circle about the space-fixed-z axis. In this case, α→α+π and β→β. The intersection line (green) will also rotate by same amount, π/2. Then, how to measure γ? From the old green line or from the rotated green line?
If the latter case is true, does this mean γ is always fixed (because it's always measured from same line)?

What if we extended the question to mirror reflection of the body and its attached-XYZ axes on a space-fixed plane, say xz? Here, α→2π-α and β→β, but how to measure γ?

 

[stevendaryl]

Thanks a lot for the reply. I do understand the definition of Euler angles. I feel like the way I put my question was not very clear so let me rephrase it:

what happens when we rotate the body itself with its attached XYZ axes about the space-fixed-z axis?
For example in the above figure, let's rotate XYZ about space-fixed-z axis by π/2. This can be viewed as rotating the blue circle about the space-fixed-z axis. In this case, α→α+π and β→β. The intersection line (green) will also rotate by same amount, π/2. Then, how to measure γ? From the old green line or from the rotated green line?
If the latter case is true, does this mean γ is always fixed (because it's always measured from same line)?

What if we extended the question to mirror reflection of the body and its attached-XYZ axes on a space-fixed plane, say xz? Here, α→2π-α and β→β, but how to measure γ?

Are you asking how to rotate the XYZ axes starting from the orientation in the picture? The usual definition of the Euler angles are in terms of a starting point in which the space-fixed x-axis lines up with the body-fixed X axis, etc.