Concept

Inversion

The map that sends each point out along its own ray from a fixed centre, to the distance whose product with the original is the square of a fixed radius. It carries circles and lines to circles and lines and preserves angles, which is what turns a hard tangency question into an easy one.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

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.

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.

geometry · inversion

The midpoint of a segment, drawn with a compass and no straightedge. A segment with the arcs that step its length three times round one end to reach the point twice as far away, and the further arcs that send that point back to the midpoint, every one of them a circle.

The straightedge buys nothing

Every point a compass and a straightedge can construct together can be constructed by the compass alone. The straightedge draws lines nobody needs; the compass does the work, and the proof that it does is an inversion performed with arcs.

computation · compass only

Named alongside it

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

Construction Circle Circle preserving Compass Conformal map Constructible number Cross ratio Incidence Midpoint Operation set Projective map Ptolemy inequality

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