A simple closed curve $\mathcal{C}$ in the plane is such that, going along the curve from a point $P$ thereon and getting back to it, the total angle has measure $2\pi$. So one can write $2\pi=\int_{\mathcal{C}} d\theta=\int_{\mathcal{C}}\dfrac{d\theta}{ds}ds=\int_{\mathcal{C}}C(s)ds$ where $C(s)$ is the curvature and $s$ the abscissa on the curve. Of course $C$ is continuous and $T$-periodic for $T=\int_{\mathcal{C}}ds$.
So let's consider the space $S(T)$ of all $T$-periodic continuous functions on $\mathbb{R}$ $f$ such that $\int_{0}^{T}f(t)dt=2\pi$.
Is there a bijection between the set of all simple closed planar curves of perimeter $T$ and $S(T)$?