Let $\gamma$ be a clesed curve in $\mathbb{R^2}$ given by $$\begin{array}{ccl} \gamma : [a,b] & \longrightarrow &\mathbb{R^2} \\ t & \longmapsto & \gamma(t)=(x(t),y(t)). \end{array}$$ locally we can parametrise the curve $\gamma$ as a graph of function $(t,f(t))$
My question is how we can express $f(t)$ in terms of $x(t)$ and $y(t)$ ?
If $x(t)$ is invertible on some interval $I\subset[a,b]$, then for all $t\in x(I)$ the point $(t,y(x^{-1}(t)))$ is on $\gamma$ and all points of $\gamma$ where $x(t)\in x(I)$ can be written in this form. Thus for $t\in x(I)$ one should define: $$f(t)=y(x^{-1}(t)).$$