Question 10.6.3: Sketching an Ellipsoid Graph the ellipsoid x²/1+ y²/4+ z²/9=...
Sketching an Ellipsoid
Graph the ellipsoid
\frac{x^2}{1}+\frac{y^2}{4}+\frac{z^2}{9}=1 .First draw the traces in the three coordinate planes. (In general, you may
need to look at the traces in planes parallel to the three coordinate planes, but the traces in the three coordinate planes will suffice, here.) In the y z \text {-plane, }, x = 0, which gives us the ellipse
\frac{y^2}{4}+\frac{z^2}{9}=1,
shown in Figure 10.56a (on the following page). Next, add to Figure 10.56a the traces in the x y \text { – and } x z \text {-planes. } These are
\frac{x^2}{1}+\frac{y^2}{4}=1 \quad \text { and } \quad \frac{x^2}{1}+\frac{z^2}{9}=1 \text {, }respectively, which are also ellipses. (See Figure 10.56b.)
\text { CASs } and many graphing calculators produce three-dimensional plots when given z as a function of x and y. For the problem at hand, notice that we can solve for z and plot the two functions z=3 \sqrt{1-x^2-\frac{y^2}{4}} \text { and } z=-3 \sqrt{1-x^2-\frac{y^2}{4}}. Such graphs often fail to connect the two halves of the ellipsoid. To correctly interpret such a graph,
you must mentally fill in the gaps. This requires an understanding of how the graph should look, which we obtained drawing Figure 10.56b.
As an alternative, many \text { CASs } enable you to graph the equation x^2+\frac{y^2}{4}+\frac{z^2}{9}=1
using implicit plot mode. In this mode, the CAS numerically solves the equation for the value of z corresponding to each one of a large number of sample values of x and y and plots the resulting points. The graph obtained in Figure 10.56c shows some details not present in Figure 10.56b, but doesn’t show the elliptical traces that we used to construct Figure 10.56b.
The best option, when available, is often a parametric plot. In three dimensions,
this involves writing each of the three variables x, y and z in terms of two parameters,
with the resulting surface produced by plotting points corresponding to a sample of values of the two parameters. (A more extensive discussion of the mathematics of parametric surfaces is given in section 11.6.) As we develop in the exercises, parametric equations for the ellipsoid are x = sin s cos t, y = 2 sin s sin t and z = 3 cos s, with the parameters taken to be in the intervals 0 \leq s \leq 2 \pi \text { and } 0 \leq t \leq 2 \pi. Notice how Figure 10.56d shows a nice smooth plot and clearly shows the elliptical traces.
