The positions of the control marks are described by coordinates based on a geodetic datum. This is a mathematical model of the Earth's shape called an ellipsoid, which looks like a sphere that has been squashed from the top and bottom.
Positions in terms of a geodetic datum are expressed as latitudes and longitudes, with units of degrees, minutes and seconds. Heights are given in metres and are measured above the surface of the ellipsoid.
The official geodetic datum in New Zealand is the New Zealand Geodetic Datum 2000 (NZGD2000). It was adopted in 1998, and the coordinates of all New Zealand's survey and control marks refer to it. NZGD2000 is compatible with the Global Positioning System (GPS), which means that positions between the two systems are equivalent.
The height or depth of a reference point is its distance above or below a reference surface or vertical datum. The reference surface normally associated with heights is the mean level of the sea.
In New Zealand, many heights are provided in relation to sea level, determined by tide gauges at different ports. Because the average level of the sea around New Zealand's coast varies by up to 0.5 metres, the heights determined in relation to the different tide gauge-based datums are offset from each other. Users need to be careful when combining heights from different datums.
The official vertical datum for New Zealand region is the New Zealand Vertical Datum 2016 (NZVD2016). It gives heights that are consistent across New Zealand and which are normally within 0.5 metres of sea level.
NZGD2000 heights are measured above the ellipsoid, which in New Zealand can be up to 35 metres different from sea level. Ellipsoid heights are also observed by GPS receivers. To change to more meaningful sea level heights, a geoid model needs to be used.
The gravity field of the Earth is not constant. It varies by small amounts due to the effects of different rock types, whether or not land is covered in water, rainfall patterns and ice coverage.
A geoid is a map of the gravity field in relation to an ellipsoid. It can be used to convert ellipsoidal heights to sea level heights.
A three-dimensional model of a geoid appears lumpy and looks a bit like a potato. In New Zealand, the official geoid is NZGeoid2016, it coincides with the mean level of the sea within 0.5 metres.
Coordinates in terms of a geodetic datum relate to a curved surface called an ellipsoid. It is not possible to represent them on a flat surface such as a map without introducing some sort of distortion. This can be visualised using the peel of an orange: it is not possible to lay the entire peel flat without breaking it in some way.
A projection is used to manage these distortions. An additional benefit of projections is that they have units of metres, which lets users measure meaningful distances directly from a map.
Many different types of projection can be used, but the most common are related to different geometric shapes such as cylinders, cones and planes. In New Zealand, most maps are produced using the Transverse Mercator projection, which is based on a cylinder that lies on its side so that it is coincident with a line of longitude (called the central meridian).
The official projection for New Zealand topographic mapping is called the New Zealand Transverse Mercator 2000 or NZTM2000.
The process of converting coordinates between datums and projections is called coordinate transformation.
When using coordinates or heights from different sources, it is important to ensure they are in terms of the same datum and/or projection. If coordinates or heights are mixed, then this can introduce errors (potentially of hundreds of metres) in the positions resulting from them.