Concept

Grid north — where it appears

The direction the grid's northing axis points, which differs from true north by the convergence angle. The difference is zero on the central meridian and grows with distance from it, and it is one of three norths a surveyor has to keep apart.

Named by 5 essays across 3 fields — each of them below, with the objects they name alongside it.

Grid north against true north at 52°, across a 6° zone. seven meridians drawn in the grid's own coordinates. The grid's north is straight up the page everywhere by construction; the meridians are not, and the angle between them is the convergence. It is exactly zero on the central meridian, reaches 2.36° — 142 minutes of arc — at 3° out, and changes sign across the middle. The manual's γ = Δλ sin φ is 3.0″ short of it at the zone edge.

Grid north is not north

A grid has one north, parallel everywhere on the sheet by construction. The Earth has a different one at every point. The angle between them reaches two degrees at the edge of a UTM zone, and a straight line on the grid is not a straight line on the ground either.

families · Ellipsoid

The order the corrections go in, and what it costs. Setting out a 51.8 km line on the UTM zone 31N means turning a grid bearing into an azimuth to observe, and there are two corrections: the meridian convergence, 0.8669° here, and the arc-to-chord correction, 6.242″. Each bar carries the lateral offset it produces at the far end of the line, because a bearing error is a number nobody can picture and a sideways miss is the thing that misses. The exact arc-to-chord and the classical formula every manual gives differ by 0.0044″, which is 1.10 mm at the far end — small, real, and the reason the corrections have an order rather than a sum.

Setting out runs the chain backwards

Turning a design coordinate into something to observe on the ground means undoing the reduction chain, and undoing a chain reverses the order as well as the operations. Two of the corrections do not commute, and getting them the wrong way round misses by a millimetre at fifty kilometres.

practice · Reduction

A parallel, and the places with the same northing as its middle. Both curves are drawn on a transverse Mercator zone 6° wide at 45° north. The upper one is the parallel — every place on it is at the same latitude — and it climbs 4379 metres of northing on the way to the zone edge. The lower one is the set of places whose northing equals the parallel's on the central meridian. Between them lies every pair the grid puts in the wrong order, and its widest point is 4.39 kilometres of latitude.

Further north on the grid is not further north

Thirteen rungs price a grid's origin, scale, units and zones, and every one treats a coordinate as a position. It is also an ordering, and the two disagree: a parallel climbs 4,379 metres of northing on its way to the edge of a UTM zone, so two places 4.4 kilometres apart in latitude can be listed in the wrong order — and the share of pairs it happens to is the convergence, in a different unit.

practice · Grid

A legend saying "illuminated from 315°" is true at one longitude. A hillshade's azimuth is measured from the top of the sheet, because the shading is computed on the projected raster. The top of the sheet is grid north, so the compass bearing the light comes from is the declared azimuth plus the meridian convergence, and that varies across the sheet. On a conic it swings by 55.3 degrees over eighty degrees of longitude. On a cylindrical projection in its normal aspect it does not swing at all, which is the flat line — the only case the legend is right everywhere.

The light comes from a page direction

A hillshade's illumination azimuth is declared from the top of the sheet, and the top of the sheet is grid north. On a conic the light therefore swings 52.6° across eighty degrees of longitude, and 72.8 per cent of the sheet is shaded differently from what the legend claims.

distortion · Gradient

Two maps with the same ellipses. Mollweide, and the same projection with the sheet turned 37° on the desk. Every ellipse is the same size and the same shape as its counterpart, and its angle to its own graticule is the same — the two indicatrix fields agree to 1.8e-12, which is arithmetic noise. The rotation differs by 37° at every point. So the classical description of a map's distortion is exactly blind to the difference between these two, and the difference is the whole of what a surveyor calls convergence.

The fourth number the ellipse does not carry

A Jacobian has four independent entries and an indicatrix reports three. The missing one is the rigid rotation in the polar decomposition A = R·S, the indicatrix is exactly S, and R is what turns north into grid north: for the transverse Mercator it agrees with the survey formula for convergence to 5 × 10⁻¹⁰ degrees, from a different library and a different derivative.

distortion · Tissot

Named alongside it

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

Convergence Transverse Mercator National Grid UTM Arc-to-chord Bearing Geodesic The (t − T) correction Verification Zone Anisotropy Aspect

All concepts