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

Nanostrain — where it appears

A part in a thousand million, which is the unit a geodetic strain rate is quoted in. A plate interior sits below one nanostrain a year and a plate boundary in the hundreds, so the unit itself separates the two cases.

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 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

Named alongside it

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

Plate motionRigid rotationStrain rateAngular deformationBaselineDatumDeformation gradientDilatationEpochFirst-order surveyJacobianPrincipal strain

All concepts