Concept

Total curvature — where it appears

The Gaussian curvature of a surface integrated over a region of it, measured in steradians. By Gauss–Bonnet it equals the region's angle excess round its boundary, so a surveyor reads it from a triangle's angles, and it exists for any region of stated size even where the curvature at a point does not.

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

The least distortion possible over a 30° region. Scale factor along a radius of the cap, each projection normalised to unit scale at the centre. Chebyshev's criterion names the projection whose scale is constant on the boundary as the conformal map of least scale variation, and for a cap that is the stereographic projection centred on it — reaching exactly sec²(ρ/2) = 1.0718 at the rim, marked. Every other conformal projection drawn here rises past that line before it gets there. It shows an optimum rather than a comparison.

Total curvature and the scale rule

The impossibility has a size. A region covering a fraction of the sphere carries a fixed amount of curvature that any flat map must absorb, and for a circular region the least distortion any conformal projection can achieve is a closed form nobody can beat.

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.

Boundary Chebyshev's bound Convergence Gauss–Bonnet theorem Gaussian curvature Geoid Lower bound Mercator National Grid Resolution Scale rule Spherical cap

All concepts