Exercise 23

Consider the ellipsoid given by the equation $$\begin{gathered} \frac{x^2}{16} + \frac{y^2}{25} + \frac{z^2}{9} = 1. \end{gathered}$$

In Activity, we found that a parameterization of the sphere $S$ of radius $R$ centered at the origin is $$\begin{gathered} x(r,s) = R\cos(s) \cos(t), \ y(s,t) = R \cos(s) \sin(t), \ \text{ and } \ z(s,t) = R\sin(s) \end{gathered}$$ for $-\frac{\pi}{2} \leq s \leq \frac{\pi}{2}$ and $0 \leq t \leq 2\pi$.

1. Let $(x,y,z)$ be a point on the ellipsoid and let $X = \frac{x}{4}$, $Y = \frac{y}{5}$, and $Z = \frac{z}{3}$. Show that $(X,Y,Z)$ lies on the sphere $S$. Hence, find a parameterization of $S$ in terms of $X$, $Y$, and $Z$ as functions of $s$ and $t$.

2. Use the result of part (a) to find a parameterization of the ellipse in terms of $x$, $y$, and $z$ as functions of $s$ and $t$. Check your parametrization by substituting $x$, $y$, and $z$ into the equation of the ellipsoid. Then check your work by plotting the surface defined by your parameterization.