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, 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?