What is the shape of an electron's circular motion in a magnetic field?

[komodekork]

So a electron moves in a circle with radius r in a magnetic field. What does that circle look like in the rest frame of the electron?

 

[DrGreg]

In the rest frame of the electron, the electron is at rest, so its path is a single point.

 

[komodekork]

So if I were to transform the coordinates of all the points on the circle in the laboratory frame by the Lorentz transformation to the rest frame of the electron, then they would all coincide with the origin of that frame?

 

[mfb]

There is no inertial frame where the electron is at rest all the time.
There is one where the electron is at rest once per revolution. In this frame, the path is a cycloid

 

[komodekork]

Maybe I should have made this more explicit. I'm talking about what would a circle with radius r in a frame S look like, in a frame S', who is the instantaneous rest frame of a particle moving on that circle in frame S. Does that make sense?

I guess my question is what would the transformation be if I want to go from the laboratory frame to a instantaneous rest frame of the electron? Just a boost parallel to the instantaneous velocity pluss a translation?
Couldn't I then just use this transformation on the parametrisation of the circle in S, to findthe corresponding curve in S'?

 

[mfb]

There is one where the electron is at rest once per revolution. In this frame, the path is a cycloid

This is the inertial frame you are looking for. It is just a boost (and I don't care about translations, as nobody defined a zero here).

 

[komodekork]

Ok, I don't get a cycloid, but this kind of curve. Does it look resonable?

edit.
I guess that's what's called an extended cycloid?