By Peano curve I mean a continuous map from the unit interval that fills the unit square in $\mathbb R^2$.

In the paper:

*Shchepin, E. V.; Bauman, K. E.*, **Minimal Peano curve**, Proc. Steklov Inst. Math. 263, 236-256 (2008); translation from Tr. Mat. Inst. Steklova 263, 251-271 (2008). ZBL1201.37061.

the authors mention the square to linear ratio, defined for a Peano curve $p:[0,1]\rightarrow[0,1]\times [0,1]$, as: $$\sup_{t_1,t_1\in[0,1]} \frac{|p(t_1)-p(t_2)|^2}{|t_1-t_2|}.$$

They prove that there exists a unique, up to isometry, regular diagonal Peano curve of fractal genus 9 that maps a unit interval onto a unit square and whose square-to-linear ratio is less than 6.

**My question is:
Are there properties such as the square-to-linear ratio that can be used to classify Peano curves up to isometry or some weaker class of transformations?**

Discrete properties that separate Peano curves into disjoint categories like whether they are self-crossing or not would also interesting.