  The signature transform, defined by the formal tensor series of global
iterated path integrals, is a homomorphism between the path space and the
tensor algebra that has been studied in geometry, control theory, number theory
as well as stochastic analysis. An elegant isometry conjecture states that the
length of a bounded variation path $\gamma$ can be recovered from the
asymptotics of its normalised signature:
$\text{Length}(\gamma)=\lim_{n\rightarrow\infty}\big\Vert
n!\int_{0<t_{1}<\cdots<t_{n}<T}d\gamma_{t_{1}}\otimes\cdots\otimes
d\gamma_{t_{n}}\big\Vert^{\frac{1}{n}}$.
  This property depends on a key topological non-degeneracy notion known as
tree-reducedness (namely, with no tree-like pieces). Existing arguments have
relied crucially on $\gamma$ having a continuous derivative under the unit
speed parametrisation. In this article, we prove the above isometry conjecture
for planar paths by assuming only local bounds on the angle of $\gamma'$ (which
ensures the absence of tree-like pieces). Our technique is based on lifting the
path onto the special linear group ${\rm SL}_{2}(\mathbb{R})$ and analysing the
behaviour of the associated angle dynamics at a microscopic level.
