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

What grid spacing a tolerance buys

Drawn here at the parameters it defaults to, with every essay that calls it.
What grid spacing a tolerance buys. The worst bilinear interpolation error across OSGB36's ground, against the spacing of the table it is interpolated from, both axes logarithmic. The smooth shift falls by exactly four for every halving, which is what second-order interpolation does. Adding a 30 centimetre ripple 2 degrees across — three parts in a thousand of the 99 metre shift it rides on — moves the spacing needed for 20 millimetres from 1° to 0.125°, which is 64 times as many nodes.

The worst bilinear interpolation error across OSGB36's ground, against the spacing of the table it is interpolated from, both axes logarithmic. The smooth shift falls by exactly four for every halving, which is what second-order interpolation does. Adding a 30 centimetre ripple 2 degrees across — three parts in a thousand of the 99 metre shift it rides on — moves the spacing needed for 20 millimetres from 1° to 0.125°, which is 64 times as many nodes.

It is drawn by datum-figure with show: "grid-spacing" — one member of a family of 11 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.

3 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.

What 4 parts per million of network strain leaves behind. Each arrow is where the best-fitting seven-parameter transformation leaves a marker, over 36 markers laid out as a grid across OSGB36's ground. The RMS residual is 1.64 metres and the worst is 3.00 metres. Seven parameters span the constant and linear parts of a displacement field; this one is quadratic, so no choice of the seven can reach it. The arrows are drawn 27× life size. What the numbers refer to

Where a fit leaves residuals

Seven parameters can carry a rigid motion and a size exactly. A triangulation network is neither, so the best possible transformation between two datums leaves metres on the table — in a pattern, not as noise — and which seven parameters come out depends on where the markers were.

 Molodensky against the exact route, NAD27. The distance on the ground between where each shortcut puts the transformed point and where the exact Cartesian route puts it, at 0 metres of ellipsoidal height, on a logarithmic scale. The full formulae stay within 0.8 centimetres across every latitude drawn; the abridged form, which replaces two ellipsoid-difference coefficients with one combined term, is worst at 40 centimetres — a factor of 48. The abridged curve dips near 45°, where the combined coefficient happens to equal the pair it replaces. What the numbers refer to

Molodensky's shortcut

The exact way to shift a datum needs an iteration nobody wanted to run on a 1950s machine, so a direct formula was derived instead. It lands within centimetres — and the abridged version everybody quotes is a different formula, worse by a factor of fifty.

 What grid spacing a tolerance buys. The worst bilinear interpolation error across OSGB36's ground, against the spacing of the table it is interpolated from, both axes logarithmic. The smooth shift falls by exactly four for every halving, which is what second-order interpolation does. Adding a 30 centimetre ripple 2 degrees across — three parts in a thousand of the 99 metre shift it rides on — moves the spacing needed for 20 millimetres from 1° to 0.125°, which is 64 times as many nodes. What the numbers refer to

When a formula is not enough

National mapping agencies distribute datum shifts as tables of numbers on a grid rather than as parameters. What decides the spacing that table needs is not the size of the shift — a hundred-metre shift tabulates coarsely, and a thirty-centimetre ripple riding on it does not.

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