Matpic
mathematics, in pictures
Concept

Ellipse

The closed conic section, and the set of points whose distances to two fixed points add to a constant. It is the cut of a cone by a plane meeting every generator, and its two foci are where the inscribed spheres touch.

Named by 12 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 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 Six points on an ellipse, and the line their opposite sides meet on. A hexagon with its six corners on an ellipse. Its three pairs of opposite sides are extended until they meet, and the three meeting points lie on one straight line. Moving one corner off the ellipse breaks the alignment.

Six points on a conic, and the line they share

Put six points on an ellipse, join them into a hexagon, and extend each pair of opposite sides until they meet. The three meeting points always lie on one straight line. The statement uses no length, no angle and no focus — which is why it holds for every conic at once, and why a straightedge alone can draw the curve through any five points.

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 Four shapes of perimeter 300 between their inner and outer circles. A square, an ellipse, a Reuleaux triangle and a stadium, each drawn with the largest circle inside it and the smallest circle around it, the ring between the two shaded, with the ring's width and the widest ring allowed.

Nearly the most means nearly round

A shape that holds almost as much as a circle of the same perimeter must almost be a circle. Bonnesen made that exact: the ring between a convex shape's largest inscribed circle and smallest enclosing circle is never wider than √(L² − 4πA)/π. Three quite different shapes holding 99% of the circle's area all have rings under 9.55 wide, and not one of 200 random convex shapes breaks the bound.

geometry · Isoperimetric 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 A disc of radius 0.2 and an ellipse of size 1.22 around the same interval. For 1/(1 + 25x²): the interval from −1 to 1 on the real axis, poles at 0 + 0.2i and 0 − 0.2i, the Taylor disc at 0 of radius 0.20, and the Bernstein ellipse with foci ±1 through the poles, with ρ = 1.2198.

An ellipse, not a disc

A Taylor series converges on a disc, and the disc's radius is the distance to the nearest singularity. Ask instead how well polynomials can follow a function on an interval, and the answer is an ellipse with the interval's ends as its foci — the largest one the function is smooth inside.

analysis · Taylor series 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 One length and three ways of guessing it. Perimeter of the ellipse with semi-axes 1, b: b=1 6.283185307, b=0.5 4.844224110, b=0.1 4.063974180, b→0 4; guesses at b=0.5: Kepler 4.442883, π(a+b) 4.712389, Euler 4.967294.

The length round an ellipse

An ellipse has an area anyone can write down, πab, and a perimeter nobody can: it is an integral with no expression in the functions of school mathematics. Kepler guessed it, Euler guessed better, and in 1914 Ramanujan wrote down a formula whose error falls like the fifth power of how far the ellipse is from a circle — off by four hundredths of a per cent even for a segment. Gauss had already found the way to compute it exactly, doubling the correct digits at every step.

analysis · Arc length

Named alongside it

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

ConicFocusHyperbolaTangencyReflectionOptimalityParabolaApproximationConvergenceConvex hullCoordinatesDandelin spheres

All concepts