Cartographic Projection
every map is wrong, and by exactly how much
Concept

Triangle inequality — where it appears

The requirement that no distance exceed the sum of the other two in any triangle. Distances measured on any surface satisfy it, which is why three places always have an exact flat picture and the trouble begins at the fourth.

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

Exact distances from two places, and from nowhere else. The two-point equidistant projection with London and Cape Town as its centres, 87.0° apart. The light circles are drawn in the map at radii of 30°, 60°, 90°, 120° about each centre; every one of them is a true distance circle on the sphere, to 1.8e-14 relative. Between two points that are not centres the drawn distance is wrong by up to 633 per cent.

A projection written as a condition

Instead of a formula, a sentence: the distance from these two places must be exactly right. The map that satisfies it is found by intersecting two circles, it is exact to five parts in a hundred million million, and it exists over the whole sphere for a reason that belongs to the sphere rather than to the construction.

choosing · Condition The four places, and every distance between them. London, New York, Tokyo, Sydney on a Mollweide projection, with every great circle between a pair drawn. The six ground distances run from 5,570 km to 16,994 km, and they are the whole of the input to every figure in this ladder — no coordinate, no projection and no coastline enters any of them. The arcs are drawn only to say which pair each number belongs to; on this page they are curves, and on the ground they are the shortest routes.

Four cities that cannot be drawn to scale

Every impossibility in this field so far has been about a surface. This one is about four numbers: London, New York, Tokyo and Sydney have six distances between them, and no four dots on any sheet of paper have those six separations. The test is a determinant Cayley wrote down in 1841, and it comes out −3.34 × 10²³ where zero is required.

impossibility · Embedding

Named alongside it

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

Closed formCayley mengerChordConditionConstraintDistance matrixEigenvalueEmbeddingEquidistanceGaussian curvatureGram matrixGreat circle

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