Ellipsoids

Let's plot ellipsoids. The first method I can think of is to slice the ellipsoid with parallel planes, as we discussed in class. So if

is our ellipsoid, then slicing with the plane gives us the ellipse

.

Observe that we only get an intersection if .

We have seen that  

,

is a parametrization for the ellipse   in the xy-plane. (This exploits the trigonometric identity .)  So we have the following parametrization for the ellipsoid:

Here's another way, which will make more sense to you in a subsequent chapter. For now, just verify that if

then

.