Why does Euler's rotational theorem require a fixed point and not a line of fixed points?

Question

If the displacement can described by a rotation through some fixed point of an object, doesn't that imply there's a line of fixed points?

Edit: The book I'm using to study is Goldstein's classical mechanics 3rd edition, the theorem goes as follows:

Euler's theorem: The general displacement of a rigid body with one point fixed is a rotation about some axis.

As far as I'm aware, the book never reflected on the consequence that there's a line of fixed points as opposed to just one fixed point when discussing the theorem, and in all applications of the theorem stated the rigid body in question had some fixed point under a displacement instead of some line of fixed points. Thanks for any help

Answer 1

The "line of fixed points" that you refer to is the axis of rotation of the object.

Reworded, the theorem says:

There is never just one fixed point in a displacement of a rigid object in three dimensions. If there is even one fixed point then there is a whole line of fixed points and the displacement is a rotation about that line, which is then called the axis of the rotation.

As with most theorems, this theorem says you have to know certain facts to apply the theorem, and then you can conclude other facts from the theorem. So here is what this theorem requires you to know:

1. You are looking at a displacement of a rigid body.
2. In that displacement, you can identify at least one point that does not move.

The theorem doesn't say you cannot use it if you know more than those two facts. It just says you do not need anything more in order to apply the theorem.

The conclusions are:

1. There not just one fixed point, but an entire line full of fixed points.
2. The displacement is a rotation.
3. The rotation rotates about that line.

How would you improve the theorem? Would you require that the person using the theorem first know that the displacement is a rotation? Would you require them to find the axis of the rotation first before they can apply the theorem? If they have to do all those things then the theorem is trivial; you would say that if you have a rotation about an axis then you have a rotation about an axis.

Since the theorem only requires you to identify one fixed point, a typical application would identify just one fixed point. Then the theorem implies that there is a whole line of fixed points. Naturally, if it is important to know which line contains all those fixed points, you need to do a little more work.