# Thread: Parametric represetation of an ellipsoid

1. ## Parametric represetation of an ellipsoid

Find the parametric representation of the ellipsoid $\displaystyle 9x^2+3y^2+4z^2=16$ using the Euler's angles. Then calculate the normal vector and the first quatratic form at (1,1,1).

Solution so far:

I use the method I learned in class and got $\displaystyle X( \phi , \theta ) = ( \frac {4}{3} sin \theta cos \phi , \frac {4}{ \sqrt {3}} sin \theta sin \phi , 2cos \theta )$

Now, the normal vector $\displaystyle n = \frac {X_{ \phi } \times X_{ \theta}} {| X_{ \phi } \times X_{ \theta }|}$

But I found that $\displaystyle X_{ \phi } \times X_{ \theta} = (- \frac {8}{ \sqrt {3}} sin^2 \theta cos \phi , \frac {8}{3} sin^2 \theta sin \phi , - \frac {32}{3 \sqrt {3}} sin \phi sin \theta cos \phi cos \theta )$, in which is getting a bit ridiculous, am I doing the right thing?

2. The z-component of your cross-product is incorrect:
$\displaystyle \mathbf{X}(\phi,\theta)=(\frac{4}{3}\sin\theta\cos \phi,\frac{4}{\sqrt{3}}\sin\theta\sin\phi,2\cos\th eta)$
$\displaystyle \mathbf{X}_{\phi}=(\frac{4}{3}\sin\theta\sin\phi,\ frac{4}{\sqrt{3}}\sin\theta\cos\phi,0)$
$\displaystyle \mathbf{X}_{\theta}=(\frac{4}{3}\cos\theta\cos\phi ,\frac{4}{\sqrt{3}}\cos\theta\sin\phi,-2\sin\theta)$
And so
$\displaystyle \mathbf{X}_{\phi}\times\mathbf{X}_{\theta}=\,$$\displaystyle \,(-\frac{8}{\sqrt{3}}\sin^2\theta\cos\phi,-\frac{8}{3}\sin^2\theta\sin\phi,{\color{red}-\frac{16}{3\sqrt{3}}\sin\theta\cos\theta\sin^2\phi-\frac{16}{3\sqrt{3}}\sin\theta\cos\theta\cos^2\phi })$
$\displaystyle \mathbf{X}_{\phi}\times\mathbf{X}_{\theta}=(-\frac{8}{\sqrt{3}}\sin^2\theta\cos\phi,-\frac{8}{3}\sin^2\theta\sin\phi,{\color{red}-\frac{16}{3\sqrt{3}}\sin\theta\cos\theta})$

That should prove much simpler, particularly when you take the norm.

--Kevin C.