Concept

Minimax — where it appears

Minimising the worst case rather than the average. It is the right objective when a claim is made about every member of a set — a scale bar says every distance is right — and its optimum equalises the error rather than concentrating it.

Named by 2 essays across 2 fields — 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. This is the only figure on the site showing an optimum rather than a comparison.

Chebyshev's criterion

The only optimality theorem in the subject. Among all conformal projections of a region, the one with least scale variation is the one whose scale factor is constant on the boundary — a criterion with a proof, a unique answer, and a test the site can run.

choosing · Choosing

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.

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.

impossibility · Embedding

Named alongside it

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

Boundary Boundary-value Cayley menger Chebyshev's criterion Closed form Convergence rate Distance matrix Embedding Estimator Gaussian curvature Laborde's projection Mercator

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