Parametrize ellipse in the xy-plane

Question

I want to parametrize the surface $(\frac{x}{a})^2 + (\frac{y}{b})^2 = 1$ in the xy-plane in $\mathbb R^3$

My attempt is $G(r,\theta) = (r \cos\theta,\frac{b}{a} \sin \theta,0)$ where $ \theta \in [0,2\pi] , r \in [0,a]$

Is my approach correct?

Answer 1

Assuming you want the disk bounded by the ellipse. The second coordinate is independent of $r$. I would suggest $$ G(r,t) = (ra\cos t, rb\sin t, 0), \quad t \in [0,2\pi), r \in [0,1] $$

But your way would also work if you multiply 2nd argument by $r$...

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If you just want the ellipse, much easier to get rid of $r$ altogether, writing $$ E(t) = (a\cos t, b \sin t, 0), \quad t \in [0,2\pi] $$

Answer 2

For completeness sake:

The canonical way to parameterize an elliptic surface is the elliptic coordianate system, i.e.

$$ x = h\cosh(\mu)\cos(\nu) \\ y = h\sinh(\mu)\sin(\nu) $$

for $\mu\in[0,\mu_0],\nu\in[0,2\pi]$. The boundary is given by the curve $\mu=\mu_0$.

In this case, for $a>b:$ $$h\cosh(\mu_0) = a \\ h\sinh(\mu_0) = b$$

This choice of parametrization is usually the most convenient one, especially if you wish to calculate integrals.