From my experience in tensor calculus, if you have a purely geometric object, then when you shift from one-coordinate frame to another, then the representation of the object changes in such a way that geometric aspect of the object is preserved under the transformation (assuming transformation is nice and invertible) .
Recently however, I ran into something quite contrary to this belief while discussing a problem of physics with a friend. The problem was as follows: consider a man standing on a moving platform spinning a yoyo tied to his finger in a vertical plane, Consider now "path" which the yoyo moves in, if we look at the path as an observer outside the platform, then we would see that the yo-yo is moving a loop-de-loop like path but the man standing on the platform sees the path of yoyo as as a circle.
Why does the path of the particle ( a purely geometric object) change as we shift reference frames? Further, how would the distortions in trajectory look like if we moved in accelerated frames of reference?
You have to transform everything that is relevant to the geometry in the same way, otherwise the geometry changes. In this case, the geometry of the path in the moving frame can be stated in the following form:
Now transforming everything into the lab frame, we actually have to transform every part of the statement into the lab frame. This includes the point $P$ (and the distance $r$ as well, but distances are invariant under Galilean transformations, so we can ignore it). If we call the transformed point $P'$, then the statement in the lab frame reads:
And this statement is still true. It's just that $P'$ is time-dependent, while $P$ wasn't.
As for the accelerated case, here is a desmos graph showing how the path would look for an accelerated platform. You can vary the acceleration yourself to see its effects.