Double Cone

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 \(A=B=1\).

x = 1.0
y = 1.0
z = 1.0

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.

A = 1.0
B = 1.0

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:

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