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

Three ways of scoring the same picture, and one exponent

Drawn here at the parameters it defaults to, with every essay that calls it.
Three ways of scoring the same picture, and one exponent. The same ladder of spans scored three ways: the worst edge of the best possible arrangement, the worst edge of the arrangement classical multidimensional scaling produces, and the root-mean-square error over all the edges. The three curves are parallel and are not the same curve — classical scaling sits a factor of 1.31 above the optimum at every span, and the root-mean-square sits below both because averaging hides the edge the minimax is about. The fitted exponents are 2.0246, 2.0223, 2.0111: the constant in front is a property of the estimator and the exponent is a property of the sphere.

The same ladder of spans scored three ways: the worst edge of the best possible arrangement, the worst edge of the arrangement classical multidimensional scaling produces, and the root-mean-square error over all the edges. The three curves are parallel and are not the same curve — classical scaling sits a factor of 1.31 above the optimum at every span, and the root-mean-square sits below both because averaging hides the edge the minimax is about. The fitted exponents are 2.0246, 2.0223, 2.0111: the constant in front is a property of the estimator and the exponent is a property of the sphere.

It is drawn by curvature-figure with show: "distance-estimators" — one member of a family of 29 figures that share a generator, so the drawing above is what that generator returns when it is asked for this one and given nothing else.

1 essay calls it. Every call below passes it something, because a placement that passes nothing draws whichever member of the family the generator happens to default to rather than the one its essay argues about.

Where it is called

Changing this changes every one of these figures.

The least error a flat picture can have, against how much sphere it spans. The same configuration of places, shrunk about its own centroid so that every bearing is kept and only the span changes, with the least worst-case relative error of the best flat picture at each size. Both axes are logarithmic. The fitted slope over the rows below ninety degrees is 2.0246: the error falls as the SQUARE of the diameter. That is Gauss's theorem arriving as a number for a finite set — curvature is a second derivative, so its first effect on a distance is quadratic in the separation — and it is why a county fits on a sheet and a hemisphere does not. The impossibility

How wrong a flat picture has to be

The rung below proves no flat picture of four places is exact and leaves the size of the failure to a determinant nobody can read. Measured directly, the least error falls as the square of how much sphere the places span — fitted exponent 2.0088 — and the same exponent comes back from five different arrangements while the constant in front of it moves by a factor of seventeen.

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