Matpic
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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 3 essays across 2 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

Named alongside it

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

ConicDandelin spheresEllipseParabolaReflectionBilliardsCausticCounterexampleDegenerate conicDirectrixEccentricityHyperbola

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