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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 $t\in(-1,1)$: an animation demonstrating the effect of these transformations on a parabola drawn on a grid

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  • $\begingroup$ Let's start in $n$-dim space $V$. If $G\subset\mathrm{Aff}(V)$ is the affine stabilizer of the subset $P$, the condition is equivalent to $\mathrm{Aff}(V)=\mathrm{Simil}(V).G$, or equivalently that $G$ acts transitively on $\mathrm{Aff}(V)/\mathrm{Simil}(V)\simeq \mathrm{PGL}(V)/\mathrm{PO}(V)$ (in particular this only depends on the linear projection of $G$). In dimension 2 this ought to be made more precise. $\endgroup$
    – YCor
    Commented Oct 31, 2020 at 18:06
  • $\begingroup$ In most interesting examples, $G$ is closed (e.g., when $P$ or its complement is closed). Then one should list closed subgroups of $\mathrm{Aff}(\mathbf{R}^2)$ with the given property and seems doable. Then for such subgroups, it consists in classifying invariant subsets. For instance in the case of the parabola $y=x^2$, what is $G$? $\endgroup$
    – YCor
    Commented Oct 31, 2020 at 18:18
  • $\begingroup$ OK, interesting. For $G$ in this particular case, $G$ has three orbits: the parabola and the two components of its complement. Here $G$ can be viewed as a lift of the lower triangular group $T$. Maybe classifying the lifts of $T$ in general can yield to other examples, or even classify them. $\endgroup$
    – YCor
    Commented Nov 1, 2020 at 19:24

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