eccentricity

Of all the numbers attached to a conic, one does the job of a single dial that turns a circle into an ellipse, then into a parabola, then into a hyperbola. That dial is the eccentricity, written e (a fixed number for each curve, not Euler's number 2.718). It measures how far a conic departs from being a perfect circle — literally, how 'off-center' it is.

There are two equivalent ways to read it. By the focus-directrix property, e is the constant ratio (distance to the focus) / (distance to the directrix) for every point on the curve. By the focal-distance picture, e = c/a, where c is the focus-to-center distance and a is the center-to-vertex distance. Either way, the value of e classifies the conic completely: e = 0 is a circle (the two foci have merged at the center); 0 < e < 1 is an ellipse (the closer e is to 0, the rounder; closer to 1, the more squashed); e = 1 is a parabola, the exact boundary; and e > 1 is a hyperbola (the larger e, the more sharply it flares open). This is the cleanest one-number classification of the whole family.

Eccentricity is the language astronomers use for orbits: Earth's orbit has e about 0.017 (nearly circular), Mars about 0.093, Halley's comet about 0.967 (a very long thin ellipse), and an object with e of 1 or more is on a one-way parabolic or hyperbolic flyby that never returns. The same single number tells you a conic's whole character before you draw a single point.