Geodesy and Geodetic Positioning
Three surfaces, one equation
Keep separate the topographic surface, the geoid (an equipotential surface approximating mean sea level) and the reference ellipsoid (a mathematical surface). The link is h (ellipsoidal height) = H (orthometric height) + N (geoid undulation). GNSS delivers h; engineering and cadastral practice need H. Ignoring N is the commonest computational error in examination questions.
Datums and transformations
A geodetic datum is an ellipsoid plus its position and orientation relative to the earth. Many national networks use a local ellipsoid with a chosen origin (Clarke 1880-based datums are widespread in former British territories), while modern positioning uses global frames such as WGS 84 and ITRF, realised at a stated epoch. Because plate motion moves points relative to the frame, coordinates must carry an epoch; comparing coordinates from different epochs without a velocity model is a standard trap. Transform with a Helmert 7-parameter similarity (three translations, three rotations, scale) or the abridged Molodensky formulae - and always confirm direction (A to B rather than B to A) and units (arc-seconds, ppm).
Projections and grid
On a Transverse Mercator grid the central meridian scale factor is 0.9996, true scale occurs on two lines roughly 180 km either side, and grid distances differ from ground distances. Combined factor = point scale factor x elevation factor (R/(R+H)). Meridian convergence must be applied to convert grid bearings to geodetic bearings.
Worked example
Given H = 240.0 m and N = -18.4 m, h = 240.0 + (-18.4) = 221.6 m. Ground distance 1000.000 m with scale factor 0.99960 and elevation factor 0.999962 gives a grid distance of 1000.000 x 0.999562 = 999.562 m, about 0.44 m shorter. Rule of thumb: 100 m of height is about 16 ppm of distance.