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Concept

Conic — where it appears

A curve cut from a cone by a plane — a circle, an ellipse, a parabola or a hyperbola, according to the angle of the cut. The kind is decided by the angle of the cut against the cone's own slope, and all four share one description in terms of a focus and a directrix.

Named by 7 essays across 3 fields — each of them below, with the objects they name alongside it.

Four conic sections from one cone. Circle, ellipse, parabola and hyperbola, produced by tilting a single cutting plane further and further.

One cone, four curves

The circle, the ellipse, the parabola and the hyperbola look like four separate objects with four separate equations. They are one object, cut at four angles.

geometry · Conic sections Rational points on the unit circle. Lines of rational slope through the left-hand point of a circle, each meeting it again at a rational point.

Every triple, on one circle

Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.

number · Pythagoras Every ray from one focus arrives at the other. An ellipse with its two foci and 13 rays leaving the first. Each is reflected at the curve by the ordinary law of reflection and each passes through the second focus.

Every ray comes back to the other focus

An ellipse has two foci and one property everybody remembers: the distances to them add to a constant. What that property forces is stranger and more useful — a mirror shaped like an ellipse sends every ray leaving one focus, in every direction, through the other.

geometry · Conic sections A room a trajectory cannot get out of, and one it can. A mushroom-shaped billiard table with two long trajectories: one confined to the cap by a conserved quantity, and one that enters the stem.

A room that cannot be lit

Mirror the walls of a room and put a lamp inside it. Every point should be lit, since light bounces forever — and there are rooms with a dark spot no ray from the lamp ever reaches.

dynamics · Billiards Every ray aimed at one focus is turned towards the other. A hyperbola with its two foci and 13 rays aimed at the far one. Each strikes the near branch from outside and is turned towards the near focus — which is the property a Cassegrain telescope's secondary mirror uses.

Aimed at one focus, turned towards the other

An ellipse sends every ray leaving one focus through the other. A hyperbola does something that sounds like the same sentence and is not — it takes a ray aimed at the far focus and turns it towards the near one, which is what the second mirror of a telescope is for.

geometry · Conic sections Two families of conics, crossing at right angles. 4 ellipses and 3 hyperbolas with the same pair of foci. Every ellipse meets every hyperbola at a right angle, checked at all 12 crossings.

Two families that cross at right angles

Fix two points and draw every ellipse with those foci, then every hyperbola with the same two. Each curve of the first family meets each of the second at a right angle, so between them they are a coordinate system — and the reason is one sentence about angle bisectors.

geometry · Conic sections One sign decides which curve it is. 3 conics drawn from the general quadratic, each labelled with its discriminant B² − 4AC and the curve that sign names, checked against how many times the curve meets a large circle.

One sign decides which curve

The general quadratic in two variables has six coefficients and draws a conic. Which of the four it draws is settled by a single combination of three of them, and the other three cannot change the answer however they are chosen.

geometry · Conic sections

Named alongside it

The objects these essays reach for when they reach for this one.

EllipseFocusHyperbolaReflectionParabolaTangencyDandelin spheresDegenerate conicEccentricityOptimalityProjectionBijection

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