Stereographic projection

Where's the maths? This is geometry — the geometry of a sphere, plus the arithmetic of scale that you use every time you read a map. A map projection is nothing more than a rule that turns a latitude φ and a longitude λ on the round Earth into a pair of coordinates on flat paper, and stereographic projection is the prettiest rule of all: stand a lamp at the North Pole and let the Earth's shadow fall on a flat table. A country lying near the table casts a shadow close to its true size, but a country creeping up towards the lamp is blown up into something monstrous — the same enlargement you learn in Sec 2, except that the scale factor changes from place to place instead of staying put. The gold circle on the globe is always the same real size, so its shadow is a direct measurement of how badly the map is lying at that spot: its area compared with the same circle at the equator is the area scale factor, and its shape tells you whether angles — and so compass bearings — survive the flattening. Switch between the three projections and you will find one that keeps shapes, one that keeps areas, and none that keeps both. That is not a failure of cleverness: Gauss proved that a sphere simply cannot be flattened, so every flat map of the world must distort something, and a cartographer's real job is choosing what to sacrifice.

Stereographic projection. Put the sphere of radius 1 with its centre at the origin and the lamp at the North Pole N = (0, 1, 0). For a point P on the sphere, follow the straight line from N through P until it hits the flat table through the equator. Writing the point in latitude and longitude, the shadow lands at distance r from the centre of the table where

The South Pole lands at the centre, the equator lands on a circle of radius 1, and the North Pole — the lamp itself — never lands at all. Two miracles follow. Angles are perfect: the gold circle stays an exact circle no matter where you put it, so the "shape" readout is 1.00 everywhere. And every circle on the sphere projects to a circle on the plane — even a great circle the size of the equator. Tick "great circle" and the app fits the best circle through the projected curve and prints how far off it is: a few parts in 10¹⁵, which is just the computer's rounding, not an error in the geometry. (Tilt the great circle until it runs through the lamp and its shadow straightens into a line — a line being a circle of infinite radius.) What stereographic destroys is size: near the lamp the area scale factor runs away to infinity.

Mercator. Wrap the sphere in a cylinder instead, and stretch the north–south direction by exactly as much as the east–west direction is already stretched. At latitude φ the parallel of latitude is a circle of radius cos φ but is drawn full width, so it has been enlarged by 1/cos φ = sec φ; matching that vertically forces

At 60° N, sec 60° = 2, so areas are multiplied by 4 — check it against the readout. Greenland really is about 2.2 million km² and Africa about 30.4 million km², a ratio of 14 to 1, yet on a Mercator map they look the same size. Mercator is not a mistake though: it was built so that a fixed compass bearing is a straight line, which is exactly what a sailor wants.

Lambert's cylindrical equal-area projection. Keep x = λ but flatten the vertical instead of stretching it: x = λ, y = sin φ Now parallels are enlarged by sec φ while meridians are squashed by cos φ, and the two scale factors multiply to sec φ × cos φ = 1. Every country is drawn with its true area — Africa is properly enormous — but the circles are squashed into ellipses, so shapes and bearings are wrong. (This is Archimedes' old result that a sphere and the cylinder around it have the same surface area, band by band.)

So: stereographic and Mercator are conformal — they keep angles and lose areas. Lambert is equal-area — it keeps areas and loses angles. The one thing no projection can do is both at once: no flat map of a sphere can keep every shape and every area right, because a sphere cannot be unrolled without stretching.

Controls: drag to orbit the globe, scroll or pinch to zoom, drag the gold circle straight across the globe, or use the sliders. Coastlines are a deliberately coarse hand-made outline (a few hundred latitude/longitude points), drawn procedurally onto both the globe and the flat map — good enough to compare sizes, not for navigation.