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Concept

Focus — where it appears

A point that a conic section is defined in terms of, and to which the curve reflects rays from its partner or from infinity. An ellipse has two of them with distances summing to a constant, a parabola one, and the reflection property follows in each case.

Named by 8 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 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 Two confocal ellipses, two confocal hyperbolas, and equal diagonals of 1.399. Two ellipses and two hyperbolas with the same foci cut out a four-sided region with curved sides. Its two diagonals, drawn as straight segments, both measure 1.3987.

Equal diagonals in a curved quadrilateral

Two ellipses and two hyperbolas sharing the same foci cut out a four-sided region with curved sides and no symmetry to speak of. Its two diagonals are nevertheless exactly equal. The reason is a stretch that carries one ellipse onto the other and moves every pair of points so that crossed distances match — a property only confocal curves have.

geometry · Conic sections The roots of the derivative inside the hull of the roots. Three polynomials of degree five, each drawn as its roots with their convex hull shaded, and the four roots of its derivative marked. Every root of the derivative lies inside the hull.

The roots of the slope stay inside

Mark the roots of a polynomial in the complex plane and stretch a band around them. However the roots are arranged, the roots of the derivative land inside the band — never outside, never on a new frontier. The reason is a balance of pushes, the same reason makes the derivative of a cubic mark the foci of an ellipse nobody asked for, and a question about how far inside the roots must sit has been open since 1958.

algebra · Polynomial roots Billiards in an ellipse: every path keeps touching one confocal conic. Two billiard trajectories in the ellipse x² + y²/0.49 = 1, one tangent to a confocal ellipse and one to a confocal hyperbola.

The caustics only an ellipse keeps

A billiard ball in an elliptical table touches the same confocal ellipse or hyperbola with every segment of its path, for ever. Bend the table by a percent and a half and most of those curves survive, but the one turning a quarter-turn per bounce breaks into a chain of four islands whose width grows like the square root of the bend. Birkhoff asked in the 1920s whether the ellipse is the only table that keeps them all; it is still not known.

geometry · Conic sections

Named alongside it

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

EllipseConicHyperbolaReflectionTangencyCoordinatesDandelin spheresEccentricityInvariantOptimalityOrthogonalityParabola

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