Concept

Arc-to-chord — where it appears

The angular correction between the direction of a geodesic on the ground and the direction of the straight line joining the same two points on a grid. It is small, it grows with the square of the line's length, and it is what a survey applies before computing a bearing on the plane.

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

Also named here as the (t − t) correction — the same set of essays touches all of them, so they are one junction rather than several.

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

Named alongside it

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

Bearing Convergence Geodesic Grid north National Grid The (t − T) correction Transverse Mercator UTM Inverse problem Scale factor Series truncation Tolerance

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