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

Dilatation — where it appears

The rate at which a piece of ground is gaining or losing area, which is the determinant of the deformation gradient minus one. It is the areal scale factor of a map of the ground onto itself, and its sign says whether the ground is opening or closing.

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 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

Named alongside it

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

Deformation gradientJacobianRigid rotationShear strainStrain rateAngular deformationAreal factorCentral differenceConformalityEqual-areaGrid coordinatesNanostrain

All concepts