Parametrization via derivatives

 Observe that 

$$\begin{equation}\frac{dx}{d\theta}=-a \sqrt{\sin(2\theta)}, \end{equation}$$

$$\begin{equation}\frac{dy}{d\theta}= b \sqrt{\cos(2\theta)},\end{equation}$$

satisfies

$$\begin{equation}\frac{1}{a^4}\left(\frac{dx}{d\theta}\right)^4 + \frac{1}{b^4}\left(\frac{dy}{d\theta}\right)^4=1.\end{equation}$$

Thus 

$$\begin{equation}x=2a E\left(\frac{\pi}{4}-\theta\ \Big| \ 2\right),\end{equation}$$

$$\begin{equation}y=2b E\left(\theta\ \big|\ 2\right).\end{equation}$$