Question 10.6.5: Sketching an Elliptic Cone Draw a graph of the quadric surfa...
Sketching an Elliptic Cone
Draw a graph of the quadric surface
x^2+\frac{y^2}{4}=z^2.
While this equation may look a lot like that of an ellipsoid, there is a
significant difference. \text { (Look where the } z^2 \text { term is!) } Again, we start by looking at the traces in the coordinate planes. For the y z \text {-plane, we have } x=0 \text { and so, } \frac{y^2}{4}=z^2 or y^2=4 z^2, \text { so that } y=\pm 2 z. That is, the trace is a pair of lines: y=2 z \text { and } y=-2 z. We show these in Figure 10.58a. Likewise, the trace in the x z \text {-plane } is a pair of lines:
x=\pm z \text {. The trace in the } x y \text {-plane } is simply the origin. (Why?) Finally, the traces in the planes z=k(k \neq 0) \text {, parallel to the } x y \text {-plane, are the ellipses } x^2+\frac{y^2}{4}=k^2 \text {. } Adding these to the drawing gives us the double-cone seen in Figure 10.58b.
Since the traces in planes parallel to the x z \text {-plane } are ellipses, we refer to this as an elliptic cone. One way to plot this with a CAS is to graph the two functions
z=\sqrt{x^2+\frac{y^2}{4}} \text { and } z=-\sqrt{x^2+\frac{y^2}{4}} \text {. In Figure 10.58c, we restrict the } z \text {-range } t to -10 \leq z \leq 10 to show the elliptical cross sections. Notice that this plot shows a gap between the two halves of the cone. If you have drawn Figure 10.58b yourself, this plotting deficiency won’t fool you. Alternatively, the parametric plot shown in Figure 10.58d, with x=\sqrt{s^2} \cos t, y=2 \sqrt{s^2} \sin t \text { and } z=s, \text { with }-5 \leq s \leq 5 and 0 \leq t \leq 2 \pi, shows the full cone with its elliptical and linear traces.
