Why Do a Planet, a Thrown Ball and a Torch Beam Share a Shape? One Cone

Take a double cone, two cones joined tip to tip and extending forever, and cut it with a flat plane. What the cut produces depends entirely on the angle of the plane relative to the cone. A plane perpendicular to the axis gives a circle. Tilting it slightly gives an ellipse, a closed oval. Tilting it until it is exactly parallel to the side of the cone gives a parabola, an open curve that never closes. Tilting it further, so that it cuts both halves of the double cone, gives a hyperbola, a pair of separate open branches. Those four exhaust the possibilities, apart from degenerate cases where the plane passes through the tip.

Each curve can be described without any cone at all, by a distance rule:

• •A circle is the points at a fixed distance from one point
• •An ellipse is the points whose distances to two points add to a constant
• •A parabola is the points equally far from a point and a line
• •A hyperbola is the points whose two distances differ by a constant
• •All four fit a single equation of the second degree in two variables
• •One number describes where each sits in the family

The reason these curves appear in physics is that a force falling off with the square of the distance produces exactly them and nothing else, which is a remarkable and not at all obvious fact. Newton proved it, showing that a body moving under gravity from a single mass must follow one of the four, with the choice determined by how fast it is going. Too slow and it follows a closed ellipse, which is a planet or a moon. Exactly at the escape speed and it follows a parabola. Faster and it follows a hyperbola, arriving from outside the system and leaving again, which is what interstellar visitors do. A thrown ball follows a small piece of an ellipse, and calling it a parabola is a simplification that treats gravity as parallel.

The curves were studied thoroughly long before anything was known that they described. Menaechmus is credited with the first investigation in the fourth century before the common era, reportedly while attacking the problem of doubling a cube. Apollonius of Perga wrote an eight-volume treatment around 200 before the common era that named the three non-circular curves and established most of their properties, and that work stood as the authority for eighteen centuries. Kepler found in 1609 that Mars moves on an ellipse, which was the first physical application of any of them. Descartes made them tractable algebraically in 1637, and Newton then showed why gravity produces exactly these and no others.

The reflective properties of the curves give them a second family of uses entirely separate from motion. A parabola reflects every ray arriving parallel to its axis to a single point, which is why dish antennas, solar concentrators and reflecting telescopes use that shape, and why a torch with a bulb at that point sends out a parallel beam. An ellipse reflects everything from one of its two special points to the other, which produces whispering galleries and is used to focus shock waves onto kidney stones. A hyperbola reflects as though rays came from the far point, which is used in the secondary mirrors of telescopes and in some navigation systems.

One double cone cut at four angles produces a circle, an ellipse, a parabola and a hyperbola, and each can be defined instead by a rule about distances. A force falling off as the square of distance produces exactly these paths and no others, with speed choosing which. Their reflective properties give dish antennas, torch beams, whispering galleries and telescope mirrors.