Why don't the angular velocity components add up for a rotation on a tilted plane?

Question

I seem to have a problem visualizing the addition of angular velocity components for rotation in a tilted plane and hoping someone will explain any basic errors I am making.

Please see diagrams below where I've drawn a circular tilted plane described by a vector R in a $3D$ $XYZ$ reference frame. Then I've drawn 3 circular projected shadows of that plane in the $XY$,$XZ$ and $YZ$ planes.

Let's assume that the vector R moves in a complete circle in 1 second, therefore the angular velocity $\omega$ (ie. 360 degrees/sec or 2π radians/secs).

I'm assuming that vector $R$ would also draw those circular shadow planes at the same time so therefore the angular velocity of the components of vector $R$ in $XY$,$XZ$,$YZ$ planes would also have an angular velocity of $\omega$.

But that doesn't make sense because if I added those component angular velocity vectors using pythagoras theorem we would have 'Angular Velocity Vector $ R' = \sqrt{\omega_x^2 + \omega_y^2+ \omega_z^2} $ and that does not equal to $\omega$.

I can't figure out what I'm doing wrong.

Answer 1

1. The projections are not circles but ellipses.
2. The projection of a vector does not rotate with uniform velocity even if the original vector does.
3. The magnitude (length) of the rotation vector designates its speed and therefore if you add up the three projected lengths using a Euclidean length then you would get

$$ \omega = \sqrt{ \omega_x^2 + \omega_y^2 + \omega_z^2 } $$

this is because the length of a vector in Euclidean space is invariant to rotations.

You can add up the areas of the projected ellipses in a vectorial way to get

$$ \text{Area of circle} = \sqrt{ {\rm A}_x^2 + {\rm A}_y^2 + {\rm A}_z^2 } $$ where ${\rm A}_i$ is the corresponding area of each ellipse.

Answer 2

I think it's my confusion regarding the definition of angular velocity. I've assumed that rate of change of angular displacement of that yellow dot around the geometric centre of an ellipse can be used as some average angular velocity = 2π rad/sec. But that is plainly wrong because angular velocity is instantaneous about a circular path.

Therefore, I cannot use the ellipse because a position vector of that yellow dot from the geometric centre of the ellipse is not necessarily a 'radius' and not always at right angles to the orbital velocity.

I am therefore misinterpreting the definition of angular velocity and applying it incorrectly to the angle traversed around a non-circular orbit in a certain period of time.

In fact, I can decompose the angular velocity (black vector) into components as shown in Diagram B.