Generator

A cone cut to give an ellipse

A generator in the geometry library, called 28 times across 6 essays. Below: what it draws at its defaults and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

conic is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

At its defaults

A cone cut to give an ellipse. A double cone intersected by a plane tilted 24 degrees from horizontal.

`show: "reflect"`

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.

`show: "tangent"`

The tangent makes the same angle with both radii — 48.7° each. An ellipse, its two foci, a point on it, the tangent there, and the two focal radii. The angle between the tangent and each radius is the same, which is the law of reflection stated as a fact about the curve.

`show: "all"`

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

`show: "directrix"`

Three curves, one rule, one number changed. A focus and a directrix, and the curves of points whose distance to the focus is a fixed multiple of their distance to the line. Below one the curve closes, at one it is a parabola, above one it has two branches.

`show: "dandelin"`

Dandelin's spheres. Two spheres inscribed in the cone, one above the cutting plane and one below, each touching the plane at a single point. Those two points are the foci of the elliptical section.

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

the two curves cross at a right angle at (0.450, 0.583) ×24
the ellipse with semi-axis 1.35 has the same foci ×7
the hyperbola with semi-axis 0.4 has the same foci ×6
every point of the 0.55 curve keeps the ratio ×3
a circle needs a cut that produces one ×1
a ellipse needs a cut that produces one ×1
a hyperbola needs a cut that produces one ×1
a parabola needs a cut that produces one ×1
and each point is as far from the focus as from the directrix ×1
and on the hyperbola ×1
and reaches it going forwards, not backwards ×1
and the two distances differ by the same amount wherever it strikes ×1
and the two legs together are the same length wherever it bounces ×1
and travels towards it rather than away from it ×1
between 3 and 60 rays are traced ×1
between one and four curves ×1
each curve is a name and six finite coefficients ×1
each ratio is between nothing and 2.4 ×1
every ellipse's semi-axis is longer than the focal distance ×1
every hyperbola's semi-axis is shorter than the focal distance ×1
every pair of curves was checked at its crossing ×1
every parallel ray reflects to the focus ×1
the crossing lies on the ellipse ×1
the curve named a hyperbola really is one ×1
the curve named a parabola really is one ×1
the curve named an ellipse really is one ×1
the cutting plane is tipped by less than a quarter turn ×1
the discriminant and the count of crossings with a large circle agree that a hyperbola is a hyperbola ×1
the discriminant and the count of crossings with a large circle agree that a parabola is a parabola ×1
the discriminant and the count of crossings with a large circle agree that an ellipse is an ellipse ×1
the ellipse has a longer axis and a shorter one ×1
the foci are between 0.2 and 3 from the centre ×1
the hyperbola has both semi-axes positive ×1
the incoming ray has a length to normalise ×1
the mirror is an ellipse, a parabola or a hyperbola ×1
the normal to the curve has a length to normalise ×1
the outgoing ray has a length to normalise ×1
the point is on the upper half of the curve ×1
the radius to the first focus has a length to normalise ×1
the radius to the second has a length to normalise ×1
the ratio is between nothing and 2.4 ×1
the reflected ray lies on the line through the near focus ×1
the reflected ray passes through the other focus ×1
the tangent direction has a length to normalise ×1
the tangent makes the same angle with both focal radii ×1
the three values give three different kinds of curve ×1
the view is one the family draws ×1
the window is between 1 and 8 wide ×1

Where it is called

Changing this generator changes every figure on this list. That is what makes the list worth publishing rather than keeping in a check script.

Geometry

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

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

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

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.

Computation

The cube that will not double

Doubling a cube needs an edge in the ratio of the cube root of two. That number satisfies an equation of degree three, three does not divide any power of two, and the oldest open problem in geometry closes in a line.

Geometry

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.

The whole library · What the figures prove