Coordinate Systems and Geodetic Computations
This domain tests the chain from field measurement to grid and geodetic coordinates: projection choice, scale and convergence factors, and forward and inverse computations. Expect numerical questions on reducing ground distances to the grid and on converting between azimuth conventions.
Projections and zones
- State Plane Coordinate System zones are conformal: Lambert Conformal Conic for zones elongated east-west, Transverse Mercator for zones elongated north-south. Conformal means local angles are preserved and scale at a point is the same in every direction.
- Universal Transverse Mercator uses 6-degree zones, a central meridian scale factor of 0.9996, and a false easting of 500,000 m.
- Grid azimuth differs from geodetic azimuth by the convergence angle, whose sign depends on which side of the central meridian the point lies. Grid north, true north, and magnetic north are three different directions.
Scale factors and reductions
- Elevation factor is approximately R divided by (R + H), where R is the mean Earth radius and H is height above the ellipsoid. Points above the ellipsoid have elevation factors below 1.
- Combined factor equals elevation factor times projection (scale) factor.
- Grid distance equals horizontal ground distance times the combined factor, to survey accuracy over ordinary project extents.
- To go from grid back to ground, divide by the combined factor. Azimuth and distance computations use latitudes and departures: delta N = d cos(azimuth), delta E = d sin(azimuth).
Worked example
A measured horizontal ground distance is 1,000.000 m at an elevation of 1,200 m above the ellipsoid, and the projection scale factor at that point is 0.9999234. Elevation factor = 6,371,000 / (6,371,000 + 1,200) = 0.9998117. Combined factor = 0.9998117 times 0.9999234 = 0.9997352. Grid distance = 1,000.000 times 0.9997352 = 999.735 m. The grid distance is about 265 mm shorter than the ground distance, which is what you expect because both factors are less than 1. For an inverse computation with delta N = -300.00 m and delta E = +500.00 m, the bearing angle is arctan(500/300) = 59 degrees 02 minutes, giving an azimuth of 180 - 59 degrees 02 minutes = 120 degrees 58 minutes.
Traps that cost marks
Multiplying by the combined factor when converting grid to ground; applying only the projection factor and forgetting elevation; reversing the convergence sign; mixing international feet with US survey feet (1 m = 3.2808333 US survey feet); assuming UTM grid north is true north; rounding scale factors to four decimals before multiplying.