Parametrize intersection of $z = \sin(x) + \sin(y)$ and $\frac{(x + 5)^2}{9} + \frac{(y − 1)^2}{16} = 1$.

Question

Parametrize the intersection of the wave surface $z = \sin(x) + \sin(y)$ with the elliptic cylinder $$\frac{(x + 5)^2}{9} + \frac{(y − 1)^2}{16} = 1$$

I'm thinking I can start with defining the second function in terms of $\cos(x)$ and $\sin(y)$, but this is difficult for me. Any help?

Answer 1

The well-known parametrization for the ellipse $\dfrac{(x + 5)^2}{9} + \dfrac{(y − 1)^2}{16} = 1$ is \begin{cases} x=-5+3\cos t,\\ y=1+4\sin t. \end{cases} so $z = \sin(-5+3\cos t) + \sin(1+4\sin t)$ shows the parametrization $$\left(-5+3\cos t, 1+4\sin t, \sin(-5+3\cos t) + \sin(1+4\sin t)\right)$$ which hasn't a nice simplification (I think)!