Slice a cone with a flat plane and the edge of the cut is a conic section. Tilt the
plane a little and you get each of the great curves of classical geometry in turn: a horizontal slice
is a circle, a gentle tilt
gives an ellipse, a slice parallel to the cone's side gives a parabola,
and a steep slice cutting both halves of the cone gives a hyperbola. One family, four
curves — and they are everywhere: planets orbit on ellipses, a thrown ball and a satellite dish trace
parabolas, and navigation systems pin your position with hyperbolas.
A single idea unifies them all — the eccentricity e, the
ratio of a point's distance from a fixed focus to its distance from a fixed line (the
directrix): e = 0 is a circle, 0 < e < 1
an ellipse, e = 1 a parabola, and e > 1 a
hyperbola.