inverse

In surveying, to inverse two points means to calculate the bearing (direction) and distance between them, given their known coordinates. This is the reverse operation of a forward calculation, where you start with a bearing and distance to find new coordinates. The inverse is fundamental whenever you need to determine what direction and how far one point lies from another on the ground.

The mathematics involves solving a spherical or plane triangle using the coordinate differences. On a plane (for small areas), you compute the horizontal distance using the Pythagorean theorem from the northing and easting differences, then find the bearing as an angle measured clockwise from north. On a sphere or ellipsoid (for larger surveys), the calculation is more complex and accounts for the curvature of the Earth. Most modern surveying software performs inverse calculations automatically, but surveyors must understand what the operation does and recognise when results are unreasonable.

Practical use and variants

Inverse calculations are used constantly in fieldwork and office work. A surveyor might inverse two known control points to check that a measured distance matches what the coordinates predict, catching errors in either the measurements or the coordinate data. When setting out construction points, an engineer might inverse from a station to each target point to determine what bearings and distances to measure. Surveyors also inverse to compute the bearing of a line or to check closure on a traverse.

The term "inverse" reflects the logical pairing with the forward problem. Where a forward calculation takes a starting point, bearing, and distance to find an ending point, the inverse takes two known points and recovers the bearing and distance. Some older texts use different language, such as "computing the line between two points" or "deriving the bearing," but inverse is now the standard term in English surveying vocabulary.