Matpic
mathematics, in pictures
Concept

Parabola

The conic section at the boundary between closed and unbounded, whose points are as far from a focus as from a line. It is the cut parallel to the cone's own slope, and every ray arriving parallel to its axis reflects through the focus.

Named by 5 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 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 Every quadratic is a point. The plane of monic quadratics x² + px + q with p across and q up. The parabola q = p²/4 divides it: the region below, shaded, holds the equations with two real roots, the curve itself the ones with a repeated root, and the region above the ones with none. 5 equations are marked and labelled.

Where two roots run into each other

Put every quadratic equation at a point of a plane, one coordinate per coefficient. Each possible root becomes a straight line there, every one of those lines touches the same parabola, and that parabola is the discriminant — the crease where the plane of roots is folded onto the plane of equations.

algebra · Completing the square Each doubling adds a quarter of what the last one added. n=6: A=2.5980762114; n=12: A=3.0000000000; n=24: A=3.1058285412; sliver ratio 0.263305.

The error that predicts itself

Archimedes' polygons close on π slowly — six doublings of the sides for two decimals. But the error does not fall at random. Each doubling adds almost exactly a quarter of the area the doubling before it added, so the rest of the slivers can be summed before they are drawn. Liu Hui saw the quarter in 263; Snell and Huygens turned it into bounds; and the 96-gon that gave Archimedes two decimals gives six once its error is known.

geometry · Circle area

Named alongside it

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

ConicEllipseDandelin spheresDegenerate conicDiscriminantFocusHyperbolaTangencyApproximationArchimedesCompleting the squareConvergence rate

All concepts