Circles, ellipses, parabolas and hyperbolas are called conic
sections. They can all be obtained by slicing the cone
$x^2+y^2=z^2$ with a plane.
Example 1: If the plane is horizontal, (say, $z=1$) then we get
a circle (say, $x^2+y^2=1$).


Example 2: If the plane is tilted less than 45 degrees,
then we get a stretched circle, better known as an ellipse.


Example 3: If the plane is tilted exactly 45 degrees (say, $z=y+1$),
then:\begin{eqnarray*}y+1 & = & z ,\cr y^2 + 2y + 1 & =
& z^2,\cr y^2 + 2y+1 & = & x^2 + y^2, \cr 2y & = &
x^2  1. \end{eqnarray*}This is a parabola.


Example 4:
If the plane is tilted more than 45 degrees, then it hits the cone
in two pieces. For instance, if we take the plane $y=1$, then
\begin{eqnarray*} z^2 & = & x^2 + y^2 ,\cr z^2 & = & x^2
+ 1 ,\cr z^2  x^2 & = & 1.\end{eqnarray*}This is
a hyperbola.


Conic sections come up throughout
science, and especially in physics and astronomy. If you throw a ball
across the room, its trajectory will be a parabola. The earth and
other planets move in ellipses around the sun. A comet that has barely
enough energy to escape from the sun's gravity will move in a
parabolic orbit, and if it has more energy than that, its orbit will
be a hyperbola.
[Note: Conic sections can be obtained by slicing any cone
with a plane, but the equations come out simplest when we use the
'standard' cone $x^2+y^2=z^2$.]
