Generator

A cone cut to give an ellipse

A generator in the geometry library, called 38 times across 8 essays. Below: what it draws with nothing chosen 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.

With nothing chosen

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

Every ray aimed at one focus is turned towards the other

The tangent makes the same angle with both radii — 48.7° each

Four conic sections from one cone

Three curves, one rule, one number changed

Two confocal ellipses, two confocal hyperbolas, and equal diagonals of 1.399

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 (0.543, 0.919) ×28
the point found by straightedge along line 1 lies on the conic through the five ×18
the ellipse with semi-axis 1.55 has the same foci ×9
the hyperbola with semi-axis 0.42 has the same foci ×8
line 1 is tangent to the ellipse ×6
the solved conic passes through point 1 ×5
every point of the 0.55 curve keeps the ratio ×3
the foci are between 0.3 and 2.5 from the centre ×2
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
all but a few of the lines give a finite construction ×1
and each point is as far from the focus as from the directrix ×1
and its sixth point lies on the conic ×1
and on its hyperbola ×1
and on the hyperbola ×1
and reaches it going forwards, not backwards ×1
and so they are for twenty-four other pairs ×1
and the image of Y on the one through Y ×1
and the other corner along the other hyperbola ×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 6 and 60 lines are constructed ×1
between one and four curves ×1
each corner lies on its ellipse ×1
each curve is a name and six finite coefficients ×1
each ratio is between nothing and 2.4 ×1
each vertex lies on the ellipse ×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
five points in general position determine one conic ×1
five points, each within four units of the centre ×1
for any two points of the first ellipse, the distances across the stretch are equal ×1
moved off the ellipse, some corner decisively breaks the alignment ×1
neighbouring tangents meet at a finite corner ×1
no three of the five points are nearly collinear ×1
sides one-two and four-five meet at a finite point ×1
sides three-four and six-one meet at a finite point ×1
sides two-three and five-six meet at a finite point ×1
six distinct points of contact, given by their angles in degrees ×1
six distinct points, given by their angles in degrees ×1
the construction fits a drawable window ×1
the crossed distances across the stretch are equal ×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 drawn construction is finite ×1
the drawn line's direction is between nought and a half turn ×1
the ellipse has a longer axis and a shorter one ×1
the ellipse's semi-axes are between 0.3 and 4 ×1
the hexagon's corners fall near enough to draw ×1
the hyperbola has both semi-axes positive ×1
the image of X lies on the confocal hyperbola through X ×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 order visits each of the six points once ×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 stretch carries a corner of the first ellipse along its hyperbola to the second ×1
the tangent direction has a length to normalise ×1
the tangent makes the same angle with both focal radii ×1
the three construction points lie on the Pascal line by construction ×1
the three diagonals pass through one point ×1
the three meeting points fall near enough to draw ×1
the three meeting points of opposite sides lie on one line ×1
the three values give three different kinds of curve ×1
the two diagonals of the curved quadrilateral are equal ×1
the view is one the family draws ×1
the window is between 1 and 8 wide ×1
tilted off the ellipse, some tangent decisively breaks the concurrency ×1
two clearly different points on the inner ellipse ×1
two ellipses, the second larger, both longer than the focal distance ×1
two hyperbolas, the second wider, both shorter than the focal distance ×1
with an outer ellipse of the same shape but different foci, the crossed distances differ ×1
with the second ellipse and hyperbola on different foci the diagonals differ ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

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

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

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.

Geometry

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.

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