I am trying to approximate the parametric equation of an ellipse with one focus at $(0,0)$. The semi-major axis, $a$, and the eccentricity, $e_c$, are known variables, with $0 \le e_c < 1$, and $a \neq 0$. I have a rough approximation of the parametric: $$\left(na^{2}\left(1-e_{c}\right)\left(\left(e_{c}-1\right)\left(e_{c}+1\right)a\right)\left(\cos\left(t\right)\right)-a\left(e_{c}\right),m\left(e_{c}+1\right)\left(1-e_{c}\right)\left(\left(e_{c}-1\right)\left(e_{c}+1\right)a\right)\cdot\sin\left(t\right)\right)$$
$m$ and $n$ are constants in terms of $a$ and $e_c$. So, what are $m$ and $n$, and if this form doesn't work, then what is the proper form? If you need reference, here is my graph.
You can use the usual parametric equation and shift $x$ to bring a focus to the origin:
$$\begin{cases}x=a\cos t-f,\\y=b\sin t\end{cases}$$ where $f$ is the half focus distance ($b=a\sqrt{1-e^2}, f=ea$).
You can also use the polar equation
$$\rho=\frac p{1-e\cos\theta}$$ ($p=\dfrac{(1-e^2)a}e$) and $$\begin{cases}x=\rho\cos\theta,\\y=\rho\sin\theta.\end{cases}$$