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

Spherical harmonics — where it appears

The functions on a sphere that play the part sines and cosines play on a line, each with a degree that sets its wavelength. A gravity field or a geoid is written as a sum of them, degree n having a wavelength of about 40,000 km divided by n, and every such series stops at a degree that decides every quantity finer than it.

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

The geoid's curvature is whatever degree the model was stopped at. The Gaussian curvature a geoid model contains, as a fraction of the sphere's own, against the degree the spherical harmonic series is truncated at. It rises in proportion to the cutoff, because a degree-n undulation moves the curvature by n(n+1) − 2 times its own amplitude while the amplitude itself falls only as n to the three halves. The dashed line is the ellipsoid's whole variation from equator to pole, 1.33 per cent; the geoid overtakes it at degree 1332, a half-wavelength of 15 km. Below that resolution the ellipsoid's flattening is the larger term and above it the geoid's own bumps are.

The geoid has a curvature only if you say where you stopped

Thirteen measurements have priced the curvature of surfaces that are stated — a sphere, an ellipsoid, a triaxial body, a surface with a hole, real relief. The surface a height actually refers to is none of them. It is an equipotential of the Earth's own gravity, it is only ever given as a series stopped at a degree, and its Gaussian curvature carries two more powers of the degree than its height does. So the omitted height converges, the omitted slope diverges as a logarithm, and the omitted curvature diverges as a power: EGM96 carries 0.36 per cent of the sphere's own curvature, EGM2008 carries 2.19, and a one-kilometre model would carry twenty.

impossibility · Curvature The geoid's curvature at a point has no limit; averaged over a triangle it settles. Dashed: the curvature a geoid model contains at a point, as a fraction of the sphere's own, rising in proportion to the degree the model stops at — from 0.36 per cent at degree 360 to 36 per cent at 36,000. Solid: the geoid's contribution to the mean curvature over equilateral triangles of 1, 10, 50 and 100 km, the rms of the anomaly in each one's spherical excess divided by that excess. Each rises while the model is coarser than the triangle and settles once it is finer: the 50 km triangle reads 0.64 per cent at EGM2008's degree 2190 and 0.68 per cent at 36,000; the 1 km triangle 2.19 per cent and 27 per cent.

A triangle reads the geoid's curvature, and that one converges

The geoid's Gaussian curvature at a point has no value: its spectrum carries two more powers of the degree than the height's, and the sum never settles. A surveyor's triangle does not measure it at a point. By Gauss–Bonnet its excess is the total curvature over its area, which on the geoid is the flux of the deflection of the vertical through its three sides — a first derivative averaged along lines, and that sum converges. A fifty-kilometre triangle reads 0.64 per cent of the sphere's curvature from the geoid at EGM2008's degree and 0.68 at degree 36,000, where the curvature at a point has grown sixteenfold. The quantity nothing could measure was the one nobody measures.

impossibility · Curvature

Named alongside it

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

Gaussian curvatureGeoidResolutionTruncationVerificationConvergenceEllipsoidEquipotentialGauss–Bonnet theoremOrthometric heightSpherical excessTolerance

All concepts