Conic sections — the circle, ellipse, parabola, and hyperbola — are the curves formed by slicing a double cone with a flat plane at different angles. Studied purely for their abstract geometric elegance by the ancient Greeks nearly 2,000 years before any practical application was known, they were later discovered by Kepler and explained by Newton to be exactly the shapes traced out by orbiting planets, comets, and satellites — one of history's most striking examples of "pure" mathematics turning out to describe physical reality.
Imagine an ice-cream-cone shape (technically, two cones joined point-to-point) and a flat knife slicing through it at different angles. Depending on the angle of the cut, you get four distinctly different curves:
In coordinate geometry, every conic section is the graph of a general second-degree equation: Ax²+Bxy+Cy²+Dx+Ey+F=0. Which specific type of conic results depends on the discriminant B²−4AC: negative gives an ellipse (or circle, if additionally A=C and B=0), zero gives a parabola, and positive gives a hyperbola.
An ellipse is the set of points where the sum of the distances to two fixed points (the foci) is constant. Standard equation (centred at origin): x²/a² + y²/b² = 1. This "sum of distances" property is exactly why an ellipse can be physically drawn using two pins and a loop of string — the classic "gardener's ellipse" construction.
A parabola is the set of points equidistant from a fixed point (the focus) and a fixed line (the directrix). Standard equation: y²=4px. Any ray travelling parallel to a parabola's axis reflects directly through the focus — precisely why parabolic mirrors and dishes are used to concentrate light, radio signals, or sound at a single point.
A hyperbola is the set of points where the difference of the distances to two foci is constant. Standard equation: x²/a² − y²/b² = 1. A hyperbola has two separate branches and approaches, but never quite touches, a pair of straight-line asymptotes as it extends to infinity.
The eccentricity e measures how much a conic deviates from being a perfect circle: e=0 for a circle, 0<e<1 for an ellipse, e=1 for a parabola, and e>1 for a hyperbola. This single elegant parameter provides a unified way of classifying and smoothly interpolating between all four conic types.
Johannes Kepler, analysing decades of remarkably precise observational data on Mars painstakingly compiled by Tycho Brahe, discovered (published 1609) that Mars's orbit is not a perfect circle (as had been assumed since antiquity, including by Copernicus) but rather an ellipse, with the Sun positioned at one focus rather than at the exact geometric centre. This was Kepler's First Law of planetary motion — a genuinely revolutionary break from over 2,000 years of the deeply entrenched assumption that celestial motion must be perfectly circular.
Isaac Newton, in the Principia Mathematica (1687), rigorously proved that Kepler's empirically-observed elliptical orbits follow as a direct mathematical consequence of his law of universal gravitation (an inverse-square force) combined with his laws of motion. This was a landmark unification: showing that the same simple mathematical law governing a falling apple on Earth also correctly and precisely governs the majestic orbits of planets around the Sun — with conic sections providing the precise geometric vocabulary connecting the two, seemingly unrelated physical phenomena.
Under simple Newtonian gravity, the trajectory of any two gravitationally-interacting bodies is always necessarily some conic section: an ellipse for a bound, permanently orbiting object (like a planet or a satellite), a parabola for an object with exactly the minimum "escape velocity" needed to permanently leave the system, and a hyperbola for an object moving fast enough to escape the system's gravity entirely and never return (typical of many comets and interstellar objects passing through, like 'Oumuamua in 2017).
Apollonius of Perga's treatise Conics (c. 200 BCE), surviving substantially complete in 7 of its original 8 books, is widely regarded by historians of mathematics as one of the greatest surviving works of Greek mathematics. Apollonius was the first to derive all four conic curves from slicing a single, common double cone at varying angles (earlier mathematicians like Menaechmus, c. 350 BCE, had studied each curve somewhat more separately, typically using differently-shaped and differently-angled cones for each individual curve type), and he also introduced the now-standard names "ellipse," "parabola," and "hyperbola" themselves.
In projective geometry (a broader geometric framework extending Euclidean geometry by systematically adding "points at infinity"), all conic sections become, in a precise technical sense, projectively equivalent to one another — the seemingly sharp distinctions between an ellipse, a parabola, and a hyperbola arise only from how the curve happens to relate to a particular, arbitrarily chosen "line at infinity," rather than reflecting any fundamentally different underlying projective structure. This deep unification, developed substantially by Jean-Victor Poncelet in the early 19th century, reveals conic sections to be, from the sufficiently general projective viewpoint, all fundamentally "the same kind of curve" in a rigorous and precise mathematical sense.