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: