parabola

A parabola is the curve you get by slicing a cone parallel to one of its sides. Equivalently, it is the set of points equidistant from a fixed point (the focus) and a fixed straight line (the directrix). In Cartesian coordinates it takes the form y = a·x² + b·x + c: a quadratic function of a single variable. The shape has a single minimum or maximum — the vertex — and a mirror-symmetry axis passing through it.

Apollonius of Perga studied parabolas as pure geometry in the third century BCE. They re-entered physics through Galileo, who showed in 1638 that the trajectory of a projectile under constant gravity is a parabola, and again through Kepler, whose analysis of orbits revealed that a parabolic path is the boundary case between a closed elliptical orbit and an open hyperbolic escape — the trajectory an object follows when it has exactly the escape energy.

Parabolas also appear in optics and engineering: a parabolic mirror focuses parallel incoming rays to a single point, which is why reflecting telescopes, satellite dishes, and solar concentrators all use parabolic surfaces. Suspension bridges hang in a cable shape that is very nearly parabolic when the roadway is uniform. The curve is the characteristic signature of any physical process where one quantity depends on the square of another.