### 1.  Mathematical Surfaces

To accurately express absolute or relative positions requires a mathematical surface. In Plane Surveying, the presumption is our piece of the world is small enough that we can use a local plane. As project extent increases, errors are introduced unless we account for the Earth's size and shape. That requires a mathematical reference model more complex than a simple plane.

That model is a datum. We have two distinct datums, one for vertical measurements (elevations), the other for horizontal. The North American Vertical Datum of 1988 (NAVD 88) was briefly described in the IV. Elevations topic. The theme of this topic are geodetic datums for horizontal positioning.

The NGS Geodetic Glossary defines a geodetic datum as:

(1) A set of constants specifying the coordinate system used for geodetic control, i.e., for calculating coordinates of points on the Earth...

(2) The datum, as defined in (1), together with the coordinate system and the set of all points and lines whose coordinates, lengths, and directions have been determined by measurement or calculation.

We'll step through the basic process of going from the irregular surface on which we measure to a mathematical one which can be used to express positions accurately. Although it is a horizontal geodetic datum, it isn't flat. A later topic will discuss how we develop a flat horizontal grid coordinate system from the 3D horizontal datum.