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

Deformation gradient — where it appears

The four partial derivatives of the map that carries the ground from where it was to where it is. It is the same kind of object as a projection's Jacobian, so the same construction reads two principal values, an areal factor and an angular deformation off it.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

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 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.

Rigid rotationStrain rateDilatationJacobianShear strainVelocity fieldAngular deformationAreal factorCentral differenceConformalityEllipsoidal heightEqual-area

All concepts