Matpic
mathematics, in pictures
Concept

Conformal

Preserving angles: two curves crossing at some angle have images crossing at the same angle. A conformal map is a similarity at each point separately, scaling by a factor that varies from place to place, so it keeps shapes small-scale and distorts sizes.

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

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 An astrolabe is a stereographic projection in brass. Latitude 52°: 16 engraved curves, worst departure from a circle 8.88e-15; altitude-circle formulas within 2.2e-16; Cancer radius 0.6564, Capricorn 1.5235 of the equator's.

An instrument that computes with circles

An astrolabe is a stereographic projection of the sky cut in brass. Because the projection sends circles to circles, every line on it can be drawn with compasses, and turning the star map over the plate is a rotation that answers questions in spherical astronomy without a single calculation. Of the lines medieval makers engraved, one family is not a circle on the sphere — and the arcs they drew for it miss the truth by a third of a millimetre.

topology · Stereographic projection

Named alongside it

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

ProjectionStereographic projectionMobius transformationRotationSphereAngleAreaAstrolabeCircleCircle-preservingComplex numbersFixed point

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