Azimuth - Calculating Azimuth

Calculating Azimuth

We are standing at latitude, longitude zero; we want to find the azimuth from our viewpoint to Point 2 at latitude, longitude L (positive eastward). We can get a fair approximation by assuming the Earth is a sphere, in which case the azimuth is given by

A better approximation assumes the Earth is a slightly-squashed sphere (a spheroid); "azimuth" then has at least two very slightly different meanings. "Normal-section azimuth" is the angle measured at our viewpoint by a theodolite whose axis is perpendicular to the surface of the spheroid; "geodetic azimuth" is the angle between north and the geodesic – that is, the shortest path on the surface of the spheroid from our viewpoint to Point 2. The difference is usually unmeasurably small; if Point 2 is not more than 100 km away the difference will not exceed 0.03 arc second.

Various websites will calculate geodetic azimuth – e.g. the NGS site. (That site is simpler than it looks at first glance; its default is the GRS80/WGS84 spheroid, which is what most people want.) Formulas for calculating geodetic azimuth are linked in the distance article.

Normal-section azimuth is simpler to calculate; Bomford says Cunningham's formula is exact for any distance. If is the reciprocal of the flattening for the chosen spheroid (e.g. 298.257223563 for WGS84) then

If = 0 then

To calculate the azimuth of the sun or a star given its declination and hour angle at our location, we modify the formula for a spherical earth. Replace with declination and longitude difference with hour angle, and change the sign (since hour angle is positive westward instead of east).