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Concept

Stereographic projection

The correspondence between a sphere with one point removed and a plane, made by shining rays from that point. It takes every circle to a circle or a line and preserves every angle exactly, which is why it is the projection navigators and crystallographers use.

Named by 3 essays across 2 fields — each of them below, with the objects they name alongside it.

Rational points on the unit circle. Lines of rational slope through the left-hand point of a circle, each meeting it again at a rational point.

Every triple, on one circle

Draw a line of rational slope through a single point of a circle. Wherever it comes out is a rational point, and clearing the denominators turns it into a Pythagorean triple — so every triple there is comes from one line through one point.

number · Pythagoras A circle stays a circle, unless it meets the pole. 3 circles on a sphere beside their stereographic images in the plane, which are circles, together with one circle through the projection point whose image is a straight line.

Angles survive and areas do not

Stereographic projection takes every circle on the sphere to a circle or a line, and every crossing angle to itself. It does both exactly, with no approximation anywhere, and it destroys area so thoroughly that a patch near the pole can be a thousand times its neighbour's size.

topology · Stereographic projection A turn of the sphere, seen from the plane. A square grid in the plane and its image under the map obtained by lifting to the sphere, rotating by 62° about a tilted axis, and coming back down. The lines become arcs of circles and the crossings stay at right angles.

The sphere that complex numbers live on

Add one point to the complex plane and it becomes a sphere. The rotations of that sphere are exactly the maps written as one linear expression divided by another, so a fact about turning a ball is a fact about dividing polynomials.

topology · Stereographic projection

Named alongside it

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

ProjectionConformalAngleAreaBijectionCircleComplex numbersConicDescentFixed pointHypotenuseInversion

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