50 Mathematical Ideas You Really Need to Know

The conic sections

Classical curves

The first family in the realm of the classical curves are what we call ‘conic
sections’. Members of this family are the circle, the ellipse, the parabola, and the
hyperbola. The conic is formed from the double cone, two ice-cream cones
joined together where one is upside down. By slicing through this with a flat
plane the curves of intersection will be a circle, an ellipse, a parabola or a
hyperbola, depending on the tilt of the slicing plane to the vertical axis of the
cone.
We can think of a conic as the projection of a circle onto a screen. The light
rays from the bulb in a cylindrical table lamp form a double light cone where the
light will throw out projections of the top and bottom circular rims. The image
on the ceiling will be a circle but if we tip the lamp, this circle will become an
ellipse. On the other hand the image against the wall will give the curve in two
parts, the hyperbola.
The conics can also be described from the way points move in the plane. This
is the ‘locus’ method loved by the Greeks, and unlike the projective definition it
involves length. If a point moves so that its distance from one fixed point is
always the same, we get a circle. If a point moves so that the sum of its distances
from two fixed points (the foci) is a constant value we get an ellipse (where the
two foci are the same, the ellipse becomes a circle). The ellipse was the key to
the motion of the planets. In 1609, the German astronomer Johannes Kepler
announced that the planets travel around the sun in ellipses, rejecting the old
idea of circular orbits.