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;

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.

Added: In the comments, YCor asks which subgroup of $\mbox{Aff}(\mathbb{R}^2)$ preserves the graph of $y=x^2$. If my calculations are correct, this group consists of the transformations of the form $$\tau(s, t) := \left(\begin{array}{cc|c}s & 0 & t \\ 2st & s^2 & t^2\end{array}\right)$$ for $s\neq 0$. Note that $\tau(s,t).\tau(s',t') = \tau(ss', t + st')$.

Here’s an animation demonstrating the effect of these transformations for $s=1$ and $t\in(-1,1)$: $s=1$ and $t\in(-1,1)$" />