# Find parametric equation of ellipse given semi-major axis, one focus at (0,0), and eccentricity

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 ($$b=a\sqrt{1-e^2}, f=ea$$).
$$\rho=\frac p{1-e\cos\theta}$$ ($$p=\dfrac{(1-e^2)a}e$$) and $$\begin{cases}x=\rho\cos\theta,\\y=\rho\sin\theta.\end{cases}$$
• Thanks, but would we have $x = a \cos t \pm f$ instead of having a $+a$ in it? – fasterthanlight Oct 18 '20 at 22:08