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;

There are also “degenerate” examples that in some sense vary in only one dimension:
* Any subset of a line;
* Any superset of the complement of a line;
* An open half-plane together with any subset of its boundary;
* The product of a line with any subset of a line.

Are there other examples? I’m especially interested in examples that don’t fall into the degenerate category.