Concept

Circle preserving

A property of a map: every circle and every straight line is carried to a circle or a straight line. Inversion has it, and it is why lines and circles are one family rather than two — a line being the circle that passes through the point at infinity.

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

One point on the sphere for every point of the plane. Lines from the north pole of a sphere through each of its points land on a plane below, matching the sphere minus one point with the whole plane.

A sphere is a plane plus one point

Remove a single point from a sphere and what is left can be flattened out to cover an infinite plane exactly. The construction is one straight line, repeated.

topology · Stereographic projection

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

Named alongside it

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

Conformal map Astrolabe Bijection Circle Construction Continuity Inversion Point at infinity Projection Ptolemy inequality Riemann sphere Similar triangles

All concepts