I aim to introduce a way to intuitively measure 'the similarity of $2$ subsets of $\mathbb{R}^2$ up to (Euclidean)-isometries and scaling'. Here is my attempt.
$\bullet$ First, we introduce the 'Scaled Hausdorff distance' as follows:
For 2 bounded subsets $A,B$ of $\mathbb{R}^2$, we define: $$ d_{SH}(A,B):=d_{H}\left(\frac{A}{\mathrm{diam}(A)},\frac{B}{\mathrm{diam}(B)}\right)$$
where $d_H$ is the usual Hausdorff distance.
$\bullet$ Next, let $E(2)$ be the $2$-dimensional Euclidean group, let $\mathcal{P}_{bdd}(\mathbb{R}^2)$ be the collection of all bounded subsets of $\mathbb{R}^2$. Then for any $f\in E(2)$, $f:\mathbb{R}^2\rightarrow \mathbb{R}^2$ induces bijection $$\begin{aligned} \tilde{f}: \mathcal{P}_{bdd}(\mathbb{R}^2)&\rightarrow \mathcal{P}_{bdd}(\mathbb{R}^2)\\ A &\mapsto \mbox{ the image of } f \mbox{ restricted to } A \end{aligned} $$ In other words, it induces a group action of $E(2)$ on $\mathcal{P}_{bdd}(\mathbb{R}^2)$. (See e.g. here) Moreover, elements in $E(2)$ are Euclidean isometries, hence are $d_{SH}$ isometries. Therefore we can define the distance on the quotient space $\mathcal{P}_{bdd}(\mathbb{R}^2)/E(2) $ as: $$ \widetilde{d_{SH}}([A],[B]):=\inf\{d_{SH}(A,B):A\in [A],B\in [B]\}$$ (See e.g. here)
And $\widetilde{d_{SH}}$ appears to be a metric measuring the similarity of $2$ subsets of $\mathbb{R}^2$ up to (Euclidean)-isometries and scaling.
Question: Are there any problems in my construction (although seems natural) ? Has this distance I defined appeared similarly in previous books/literature, or have any analogous definitions been established?
Edit: Here is the reason why I introduce all of these.
By introducing these definitions, I want to formalize the following intuition: Let $F:\mathbb{R}^2\rightarrow \mathbb{R}^2$ be a $C^2$ regular map, let $B_i:=[-\frac{1}{i},\frac{1}{i}]^2$ be a sequence of boxes shrinking to the origin. I want to formalize that $f(B_i)$ 'tends to' the parallelogram generated by the vectors $\frac{\partial F}{\partial x_1}(0,0)$ and $\frac{\partial F}{\partial x_2}(0,0)$.
So are there already any formulations of this statement, maybe without of Hausdorff-like distance?
Thanks for your help!
