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

A coordinate with no date on it

Drawn here at the parameters it defaults to, with every essay that calls it.
A coordinate with no date on it. Ground displacement against elapsed time for five places, each on its own plate, computed from the plate's rotation vector. Honolulu moves 71 millimetres a year and reaches 2.85 metres in 40. The dashed line is 10 centimetres, which is the tolerance an ordinary boundary survey works to: every one of these crosses it, and three of them cross it within five years.

Ground displacement against elapsed time for five places, each on its own plate, computed from the plate's rotation vector. Honolulu moves 71 millimetres a year and reaches 2.85 metres in 40. The dashed line is 10 centimetres, which is the tolerance an ordinary boundary survey works to: every one of these crosses it, and three of them cross it within five years.

It is drawn by datum-figure with show: "epoch-drift" — 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.

5 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 same coordinate on four datums. One pair of numbers — 2.0° west, 54.5° north — read as a coordinate on OSGB36, ED50, NAD27 and on WGS84, and plotted where each reading puts the mark on the ground. The spread runs to 195 metres. The numbers are identical; only what they refer to differs. What the numbers refer to

What a coordinate refers to

A latitude and a longitude are not a place. They are a claim about a model, and the model has a shape, a position, a set of marked stones and a date — change any one of the four and the ground under the numbers moves by hundreds of metres.

 Where the ground is going, six plates. The velocity of the ground under a coordinate, computed as the cross product of each plate's rotation vector with the position, and drawn along the band where that rotation is fastest rather than over an outline this site does not hold. The quickest arrow is 75 millimetres a year on the Pacific plate, which is 1.89 metres in twenty-five years. drawn in Robinson — the arrows' directions are the projection's as much as the plates'. What the numbers refer to

The epoch is part of the coordinate

A latitude and a longitude with no date on them are incomplete, because the ground they refer to is moving at between ten and seventy millimetres a year. Australia crosses an ordinary survey tolerance in under two years and moved 1.8 metres between two versions of its own datum.

 The same coordinate on four datums. One pair of numbers — 2.0° west, 54.5° north — read as a coordinate on OSGB36, ED50, NAD27 and on WGS84, and plotted where each reading puts the mark on the ground. The spread runs to 195 metres, and the height changes by 51, 44, 47 metres, which is the component nobody quotes. What the numbers refer to

The third coordinate moves too

A datum shift is quoted as a horizontal displacement because horizontal is what people look at. The transformation acts on a three-dimensional point, and its vertical component is between a quarter and a half of the horizontal one — 51 metres, for a British coordinate.

 A published coordinate is a definition being quoted. Re-observing a control point perfectly, with an instrument good to 20 mm, and comparing the answer to what is published for it. The published coordinate was fixed on a datum realisation that has since been superseded, which accounts for 106 m, and the ground it marks has moved 750 mm in 30 years at 25 mm a year. Both numbers are larger than the observation and neither is an error in it: subtracting a published coordinate from an observed one measures the interval between two definitions. Grids, and what a survey does

A published coordinate is a result

Re-observe a control mark perfectly, with an instrument good to twenty millimetres, and the answer disagrees with the published value by a hundred metres. Neither number is wrong. The difference measures the interval between two definitions.

 An error in a rotation rate is a longitude error that never stops growing. Where the prime meridian of each body has got to, if its published rotation rate is wrong by one unit in its last published decimal. Every line is straight through the origin, because the error is the rate error multiplied by the elapsed time and nothing else — there is no date after which it settles. Jupiter reaches 2279 metres at the equator after a century; Mars, whose rate is published to twelve decimals rather than seven, reaches 0.0011 metres over the same interval. What the numbers refer to

A longitude that drifts with the rotation rate

Ten essays here map bodies whose shape is the problem. A longitude is not about shape: it is a landmark plus an extrapolation over however many days have passed, and an error in the last published decimal of a rotation rate is a coordinate error that grows without bound in time.

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