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.
Best Answer
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 $$\begin{cases}x=\rho\cos\theta,\\y=\rho\sin\theta.\end{cases}$$