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

Rigid rotation — where it appears

A motion of the whole sphere about an axis through its centre, which moves every point and changes no distance between any two. It is the motion plate kinematics assigns to a plate, and it is the one motion with no deformation in it at all.

Named by 6 essays across one field — each of them below, with the objects they name alongside it.

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.

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.

datums · Bodies a transform boundary: what the ground is doing to itself. The same 72 places as the velocity field, with the strain rate computed from the motion's own four partial derivatives rather than from its size. Each cross carries two principal rates: the long stroke is the greater extension, the barred one is shortening. The largest second invariant anywhere here is 241.0 nanostrain/yr, and the arithmetic is the same arithmetic that reads a projection's indicatrix. Drawn in Azimuthal equidistant centred on the window.

The ground has an indicatrix too

Two hundred and forty-six essays hold the Earth still while the page is measured. The ground is moving at tens of millimetres a year, and the motion is a map with four partial derivatives — so the same construction that draws Tissot's ellipse draws one for the ground, and the fastest plate in the model returns exactly nothing.

datums · Strain One velocity field, three frames. a transform boundary, drawn three times. The ground is the same ground and the arrows are not the same arrows: the fastest velocity anywhere in the 3 panels is 70.6 mm/yr and in the quietest panel the same places are nearly still. A frame is a choice of which rotation to subtract, and there is no measurement that picks one. Every panel has the same strain rate at every point.

A velocity needs a frame and a strain rate does not

One place on a plate boundary moves at 5.3, 7.0, 10.2 or 54.6 millimetres a year, and towards the south-east or the north-west, depending on which rotation was subtracted first. Every one of those readings reports the same strain rate, to two parts in a hundred thousand million.

datums · Strain How long a network has before it breaks its own tolerance. The years until the worst baseline in a 400-kilometre network exceeds five millimetres plus one part per million of its length, which is an ordinary first-order specification. A boundary zone breaks it in 16 years; a plate interior takes 1819; a rigid plate never does, at any speed, because a rigid body keeps every distance it has. Bars are clipped at five thousand years.

A grid stops fitting the ground it was laid on

A hundred-kilometre baseline across a plate boundary changes by 815 millimetres in fifty years, and the same baseline on the fastest plate in the model changes by nothing at all in the same fifty. The interval before an ordinary first-order specification is broken is six years in one place and never in the other.

datums · Strain A ground that is not deforming, read off a Mercator sheet. Every place on this map is carried by a rigid rotation of the whole Earth at forty millimetres a year — the motion that deforms nothing, and that this collection has already shown deforms nothing. The circles are the strain rate a geodesist would report from the grid coordinates alone, up to 17.4 nanostrain/yr. None of it is on the ground. It is the projection's own scale factor changing along the displacement, which is a second derivative of the map arriving in a first-order measurement.

The strain a map adds to the ground's

A ground carried rigidly at forty millimetres a year deforms nothing, and read off a Mercator sheet at sixty degrees north it reports 10.9 nanostrain a year. Along a profile through a real boundary the invented part is 0.0 per cent of the answer where the zone is loud and 884 per cent where it is quiet.

datums · Strain A compacting basin, and the tilt it produces. The stated vertical velocity field — a bowl of subsidence 120 km across, with no mass leaving — drawn as circles proportional to the rate, with the tilt of the ground surface as the arrows. The subsidence is largest at the centre and the tilt is exactly zero there, because a smooth bowl has no gradient at its own bottom. The largest tilt is 125.5 nanoradians a year, on a ring at the bowl's own scale length over root two, and it is the quantity a levelling network measures.

A vertical rate needs a height system

Four rungs of this anchor measure the two-by-two horizontal tensor, because that is what a tangent chart returns. The larger signal in a subsiding basin is vertical — 126 nanoradians a year of tilt against 0.76 nanostrain a year of horizontal strain — and it is not a measurement at all until the surface it is measured against is named, because that surface is moving too.

datums · Strain

Named alongside it

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

Strain rateDeformation gradientEpochPlate motionReference frameShear strainVelocity fieldDilatationJacobianNanostrainPrincipal strainRealisation

All concepts