A parabola P in the plane has the nice property that the image of P under any affine transformation is similar to P itself.

**Which other subsets of the plane have this property?**

I wondered aloud about this [on Twitter](https://twitter.com/robinhouston/status/1322491495627268097?s=20), where Zeno Rogue gave some additional examples:
* The complement of a parabola;
* One connected component of the complement of a parabola;
* Any subset of a line;
* Any superset of the complement of a line;
* An open half-plane together with any subset of its boundary.

Are there other examples?