Concept

Azimuth — where it appears

A direction measured as an angle clockwise from north, which on a map means clockwise from whichever north the page uses. Grid north and true north differ by the meridian convergence, so an azimuth read off a sheet and an azimuth measured on the ground are two quantities with one name.

Named by 2 essays across 2 fields — each of them below, with the objects they name alongside it.

A map for pointing, not for locating. Craig's retroazimuthal projection, centred on Mecca. The straight line from any point to the centre makes an angle with the map's vertical equal to the true initial bearing from that place to the centre — checked here at 169 points, worst disagreement 5.7e-14 degrees. The bearings printed beside each line are computed on the sphere with no projection in them.

A map that cannot be read backwards

Craig's projection answers one question exactly — lay a straight edge from any place to the centre and read the compass course, right to 6 × 10⁻¹⁴ of a degree. It pays by folding: 78°S and 48°S on the same meridian are drawn at the same point, so no inverse exists and nothing else can be read off it at all.

choosing · Condition

There is no such thing as the radius of the Earth at a latitude. The two principal radii of curvature of WGS84, against latitude, with the mean radius that every table prints drawn across them. The meridional radius M runs from 6335.44 km at the equator to 6399.59 at the pole; the prime-vertical radius N from 6378.14 to the same value. They differ by 42.70 km at the equator and meet only at the pole, and the single number 6,371 km lies between them at no latitude where either is right.

The radius of curvature is two numbers

Twelve essays work on the ellipsoid and every one of them takes a radius when it needs one. There are two at every point, they differ by 42.70 kilometres at the equator, and the single number every table prints is out by 5,583 parts per million on a line running north.

datums · Ellipsoid

Named alongside it

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

Authalic radius Condition Constraint Ellipsoid Eulers theorem Flattening Gaussian curvature Gaussian radius Great circle Injectivity Inverse problem Mean radius

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