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

The scale factor across Europe, from the middle outwards

Drawn here at the parameters it defaults to, with every essay that calls it.
The scale factor across Europe, from the middle outwards. Each band is the range the larger principal scale factor takes on the ring at that distance from the centre of Europe: 0 is the middle, 1 the frontier. two of the three projections drawn are conformal, and every one of those reaches its maximum at the right-hand edge, because the logarithm of a conformal map's scale factor is subharmonic. None of them reaches a maximum inside the region.

Each band is the range the larger principal scale factor takes on the ring at that distance from the centre of Europe: 0 is the middle, 1 the frontier. two of the three projections drawn are conformal, and every one of those reaches its maximum at the right-hand edge, because the logarithm of a conformal map's scale factor is subharmonic. None of them reaches a maximum inside the region.

It is drawn by region-figure with show: "boundary-extremes" — one member of a family of 7 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.

8 essays call 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 scale factor was chosen, and the choice is a V. The worst departure of the grid scale factor from unity across a region 9° wide and 9° tall, for every scale factor between 0.998108 and 1.0002, on a central meridian of 2°W. A tangent projection touches at one meridian and is too big everywhere else, at 1987 ppm. Scaling the whole grid down slides that interval until it straddles unity, and the minimum sits at 1/√k_max = 0.999008102, where the worst departure is 993 ppm — a factor of 2.00, which is the most this construction can buy and is reached exactly. The published value marked beside it was chosen the same way, for this region, in the 1930s. Grids, and what a survey does

The scale factor was chosen

A grid's scale factor is not a constant of nature. Britain's 0.9996012717 is the reciprocal square root of the worst distortion a tangent projection would have had over the country, it halves the worst-case error exactly, and the halving is the most that construction can ever buy.

 The same six projections, ranked over Europe and Chile. Each column orders the projections by Kavrayskiy's regional criterion — the root-mean-square departure of the two principal scales from unity, integrated over the region with the area element. The lines cross, which is the point: Lambert conformal conic leads over Europe and comes fifth over Chile. A table of projections ordered by distortion is a table about somebody's region. Measuring distortion

Distortion over a region

An indicatrix describes a point and every practical question is about a country. Going from one to the other means integrating, integrating means choosing a weighting, and the weighting is the step that turns a measurement into somebody's opinion.

 What a grid of one's own is worth. For every candidate central meridian, the best worst-case distortion achievable over the region once the scale factor has also been optimised — so each point on the curve is already the bottom of its own V. The minimum is at 2.9°W with k₀ = 0.9993009, giving 700 ppm. The dashed line is UTM zone 30, whose meridian and scale factor were chosen for no region in particular, at 1059 ppm. The national grid is better by a factor of 1.51 — which is the whole of the answer to why a country publishes a grid rather than using the zones, and it is a smaller factor than the argument is usually made to sound. Grids, and what a survey does

Designing a grid for one region

A country with a grid of its own beats the international zone that would otherwise carry it by a factor of 1.51. That is worth having and it is much less than the argument is usually made to sound, and the difference between the two numbers is the whole case for and against national grids.

 The scale factor across Europe, from the middle outwards. Each band is the range the larger principal scale factor takes on the ring at that distance from the centre of Europe: 0 is the middle, 1 the frontier. two of the three projections drawn are conformal, and every one of those reaches its maximum at the right-hand edge, because the logarithm of a conformal map's scale factor is subharmonic. None of them reaches a maximum inside the region. Measuring distortion

Where the worst point is

The largest scale error on a conformal map of a country is always on the frontier, never inside it, whatever the country's shape and whichever conformal projection was chosen. It is a theorem rather than a tendency, and it is the reason Chebyshev's criterion works.

 The equal-area conic, from cylinder to plane. The largest distance between the conic at cone constant n and each of its two limits, over a shared grid, with the free scale and offset removed. Both fall as the FIRST power of the distance from their end — the slope on these axes is one — so the conic is never nearly cylindrical: halving n only halves the difference. At n = 0.999999 the conic is the Lambert azimuthal to 1.3e-6, and at n = 0.000001 it is the Lambert cylindrical to 1.2e-6. The families

The conic is the whole family

Cylindrical and azimuthal are usually presented as two of three families beside the conic. They are the two ends of it. One parameter runs from the cylinder to the plane, and both limits are exact rather than suggestive — which is measurable, and measured here.

 Sheets for a tolerance, from Chebyshev's bound and a covering. For each stated tolerance on the scale error, the cap radius at which the best possible conformal projection just meets it — sec²(ρ/2) − 1 = tolerance, which is Chebyshev's bound and has no fitting in it — and then the number of such caps needed to cover the sphere at the packing density a real arrangement achieves. One part in a thousand costs 1210 sheets of 403 kilometres radius. The slope is -0.989: a factor of ten in what the job will accept is a factor of ten in the atlas. What each projection optimises

How many sheets an atlas needs

A tolerance on the scale error inverts, through Chebyshev's bound, into a sheet radius — and a covering problem turns the radius into a count. One part in a thousand costs 1,210 sheets of 403 kilometres radius, the count goes as the reciprocal of the tolerance exactly, and the projection multiplies it by anything from one to fifty-six.

 The minimum-distortion conformal map of an elongated region, 30° by 10°. The conformal projection of an elongated region, 30° by 10° whose scale is constant on the boundary, which is Chebyshev's criterion, obtained by fitting eight terms of a series rather than by choosing a named projection. The scale factor runs from 0.98640 to 1.00000, a spread of 1.01379; on the boundary itself the largest departure from constancy is 2.13e-7 in the log, which is what the fit achieved and not what it was told. Each dot is an interior sample shaded by its own departure from the boundary's scale. What each projection optimises

Solving for the map instead of choosing it

Chebyshev's criterion has sat on this site since its second phase with one case it could be applied to: the spherical cap, whose answer is the stereographic projection. For any other region the site stated the criterion and stopped. It is a linear least-squares fit, and the fitted map beats every named projection over the region it was fitted to.

 The scale spread of every grid Britain could have adopted. A grid is conformal by requirement, so its angular deformation is zero everywhere and the whole design problem is the spread of its one remaining number: the largest scale factor over the region divided by the smallest, in parts per million. The grid's own scale factor does not enter — multiplying every scale by a constant leaves the ratio alone, which is why it is chosen last. The bottom bar is Chebyshev's optimum, the conformal map of this region whose scale is constant on its boundary, which no map of any family can beat; the adopted grid sits 1.98 times above it. Measured over a stated box rather than a coastline, because a coastline would put the vendor's generalisation into the answer. Grids, and what a survey does

The best grid a country could have had

A national grid is a conformal map chosen for one region, so its whole design problem is one number: the spread of its scale factor. That number has a theoretical floor, this site can now compute it, and the adopted grid turns out to be either exactly optimal or half as good again — depending entirely on which box the country is declared to be.

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