Generator

inversion

A generator in the geometry library, called 6 times across 1 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.

inversion 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

Inversion in a circle of radius 1. Three points and their images under inversion in a circle: each image lies on the same ray from the centre, at the distance whose product with the original is the squared radius. Beside it, the tangent construction that finds the image with compass and straightedge.

`show: "circles"`

What inversion does to circles and to lines. Three panels: a circle away from the centre inverting to another circle, a circle through the centre inverting to a straight line, and a straight line inverting to a circle through the centre.

`show: "angles"`

Inversion keeps the angle between two curves. Two crossing circles and their images under inversion. The angle at which the images cross is the same as the angle at which the sources cross, measured from the tangent directions.

`show: "chain"`

A ring of 6 circles touching two others and each other. A Steiner chain: 6 circles, each tangent to its two neighbours and to both of two nested circles, so that the ring closes. A second ring started at a different angle closes too.

`show: "ptolemy"`

Ptolemy's identity, and the line it comes from. Four points and the three products of opposite distances between them, beside the same points inverted about one of them, where the other three become collinear and the products become an addition of lengths.

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.

ring 1: and the circle it surrounds from outside ×2
ring 1: and touches the containing circle from inside ×2
ring 1: each circle touches the next ×2
a circle missing the centre inverts to a circle ×1
a circle orthogonal to the mirror is carried to itself ×1
a circle through the centre inverts to a line ×1
a point inside goes outside and a point outside comes in ×1
a point on the mirror circle is left exactly where it is ×1
and a line missing the centre inverts to a circle ×1
and every point of the straight line lands on its circle ×1
and the two short gaps overshoot ×1
between two and four points are inverted, none of them at the centre ×1
both images are circles here ×1
centre and all ×1
each image gap is the original gap divided by the two distances from the centre ×1
every point of the circle lands on its image ×1
every point of the circle through the centre lands on its image ×1
four points, in order round the circle and none of them nearly coincident ×1
neither boundary passes through the inversion centre ×1
no chain circle passes through the inversion centre ×1
no point being inverted sits at the centre ×1
off the circle the identity becomes a strict inequality ×1
off the circle the three images are not collinear ×1
on a circle the products satisfy Ptolemy's identity exactly ×1
so the two short image gaps add to the long one ×1
the angle between the two curves is the same after the map as before ×1
the chord's end sits on the mirror ×1
the concentric ring closes exactly when the radius ratio is the sine ×1
the constructed point is the one inside the mirror ×1
the drawn ring is genuinely lopsided rather than a disguised concentric one ×1
the fourth point is pushed off the circle by less than half a radius ×1
the image of the centre is not the centre of the image ×1
the image of the crossing point is on both image circles ×1
the inner circle has a sensible radius ×1
the inversion centre is inside the inner circle and off its centre ×1
the line inverted does not pass through the centre ×1
the mirror circle has a sensible radius ×1
the ring has between three and twelve circles ×1
the tangent construction lands on the same point the formula gives ×1
the three images fall on one straight line ×1
the two circles really cross ×1
the two distances from the centre multiply to the square of the radius ×1
the view is one the family draws ×1
which passes through the centre of the mirror ×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

The map that trades circles for lines

Send every point to the one on the same ray whose distance multiplies with it to a fixed number, and circles become lines, lines become circles, angles survive untouched, and a ring of tangent circles falls out of a ring of equal ones.

The whole library · What the figures prove