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

Principal strain — where it appears

The two extreme rates of extension at a place, and the directions they act along. They are the eigenvalues of the strain-rate tensor, their sum is the dilatation and their difference is the maximum shear.

Named by 2 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 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

Named alongside it

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

Plate motionRigid rotationShear strainStrain rateVelocity fieldAngular deformationDeformation gradientDilatationEpochEuler poleInvariantJacobian

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