The basic **double cone** is given by the equation
$$z^2=Ax^2+By^2$$
The double cone is a very important quadric surface, if for no other
reason than the fact that it’s used to define the so-called conics —
ellipses, hyperbolas, and parabolas — all of which can be created as
the intersection of a plane and a double cone. See Calculus textbook
for pictures of this.

The vertical cross sections of a double cone are hyperbolas, while the horizontal cross sections are ellipses. The picture below shows the cone where \(A=B=1\).

Look back at the equation for the double cone:
$$z^2=Ax^2+By^2$$
Sometimes we manipulate it to get a *single* cone. If we
solve for \(z\), we end up with a plus/minus sign in front of
a square root. The positive square root represents the top of
the cone; the negative square root gives you an equation for
the bottom. Be careful if somebody says “cone”
without elaborating; they might mean “double cone”
or “single cone”, depending on the context.

Roughly speaking, the constants \(A\) and \(B\) determine how “steep” the cone is in the \(x\) and \(y\) direction, respectively. You can get a feel from this in the second picture, which shows the portion of a cone for which \(x\) and \(y\) are between \(-1\) and \(1\). You can also change the domain to a disk, which will show you the portion of the cone for which \(-2 \leq z \leq 2\).

After you’ve played around with the examples for awhile, see if you can answer the following questions:

- Why aren’t any of the vertical or horizontal cross sections parabolas?
- Explain what happens when either \(A=0\) or \(B=0\). Why don’t you get a cone?
- Similarly, what are the cross sections given by \(x=0\) or \(y=0\)? Are these hyperbolas?