A bearing is meaningless until you say which north it is referred to, because there are three of them and they do not agree. Grid north is the direction of the northing axis on your coordinate grid. Geodetic north is the direction to the pole along the meridian through your point. Magnetic north is where a compass points.
The angle between grid north and geodetic north is the convergence of meridians, and unlike magnetic declination it is a purely geometric quantity you can compute exactly from your position and the projection definition.
Why convergence exists
Meridians on the Earth converge toward the poles — that is what a meridian is. A projection grid, by contrast, has parallel straight northing lines everywhere. The two can agree along one line, and the projection is set up so that they agree along the central meridian of the zone. Away from it they must differ, and the difference grows with distance from the central meridian.
- Convergence of meridians (γ)
- The angle at a point between grid north and geodetic north. Zero on the central meridian of the zone, positive to the east and negative to the west, and increasing with latitude.
γ ≈ Δλ × sin φThe sine of latitude in that formula is the reason convergence is a bigger deal in the north than in the south. At the equator meridians are parallel and convergence vanishes. At 45° a degree of longitude difference produces about 0.707° of convergence, and it keeps growing toward the pole.
How large does it get?
| Δλ from central meridian | Latitude | Convergence in degrees | Convergence in DMS |
|---|---|---|---|
| 0.5° east | 35° | 0.286788 | 0° 17′ 12.4″ |
| 1.5° east | 40° | 0.964181 | 0° 57′ 51.1″ |
| 2.0° east | 45° | 1.414214 | 1° 24′ 51.2″ |
| 1.25° west | 44° | −0.868323 | −0° 52′ 06.0″ |
A degree of arc is a large angle in survey terms. At 1° 24′ 51″ of convergence, a line run 1000 ft on a grid bearing lands about 24.7 ft away from where the same geodetic bearing would put it. Convergence is not a refinement; it is a first-order effect on any project that reports directions.
The sign convention people reverse
This is the single most reversed relationship in the topic, and the reason is that it is easy to reason about it backwards — thinking about where the grid lines point rather than where north lies relative to them. The reliable check is geometric rather than algebraic: sketch the meridian and the grid line at your point and read the sign off the drawing. East of the central meridian, the meridian leans toward the central meridian, which means geodetic north lies clockwise from grid north.
geodetic azimuth = grid azimuth + γ grid azimuth = geodetic azimuth − γThe third north: magnetic
Magnetic north is a different kind of quantity entirely. It is where the geomagnetic field points at your location, it changes with time, and it is affected by local iron, power lines and geology. The angle between magnetic north and geodetic north is the magnetic declination, and it has nothing to do with convergence.
Declination comes from a geomagnetic field model — in the United States, the World Magnetic Model and the higher-resolution national models published by NOAA's National Centers for Environmental Information. It must be evaluated for a specific date, because it drifts by several minutes of arc per year in many places.
| North | Defined by | Varies with | How you get it |
|---|---|---|---|
| Grid north | The northing axis of the projection | Nothing — constant across the zone | It is the grid, by definition |
| Geodetic north | The meridian through the point | Position | Grid north plus convergence |
| Magnetic north | The geomagnetic field | Position, time, and local iron | A published field model for a stated date |
The historical relevance of magnetic declination is enormous, because old deeds and original PLSS surveys were run with a compass. Retracing them requires estimating the declination at the time and place of the original survey, and that is a research problem as much as a computational one. Modern work should never establish a basis of bearings magnetically.
Basis of bearings, in practice
- State the basis of bearings explicitly on every plat, in words: grid north, a named zone and datum, or geodetic north, or a record bearing between two identified monuments.
- If reporting grid bearings, say so. A bearing labelled only N 45° 00′ 00″ E is ambiguous by up to a degree or two.
- Compute convergence rather than estimating it. It is a closed calculation from position and zone parameters.
- Never mix grid and geodetic bearings within one description. Convert all of them into one system first.
- When retracing old work, record the declination you assumed and the source, so the next surveyor can follow your reasoning.
- Remember convergence is zero only on the central meridian of the zone, so a project near a zone edge carries a large and non-negligible value.
One further subtlety worth knowing about: on a projection, the straight line between two grid coordinates is not quite the image of the geodetic line between the two points. The tiny angular difference is the arc-to-chord correction, sometimes called the second term or t-T. It is well below a second of arc on ordinary survey lines and is neglected outside precise geodetic work — but it is a real, separate effect from convergence, and it is worth knowing that the two are not the same thing.