page_title: Azimuth
section_title: Calculating azimuth
hierarchical_section_title: Azimuth / Calculating azimuth
caption_attribution_description: English: The bearing and azimuth between Cape Town and Melbourne along the geodesic (the shortest route) change from 141° to 42°. Azimuthal orthographic projection (the upper map) and Miller cylindrical projection (the lower map). The map was created using the Generic Mapping Tools, GMT, version 5.1.1.
context_page_description: An azimuth is an angular measurement in a spherical coordinate system. The vector from an observer to a point of interest is projected perpendicularly onto a reference plane; the angle between the projected vector and a reference vector on the reference plane is called the azimuth.
  When used as a celestial coordinate, the azimuth is the horizontal direction of a star or other astronomical object in the sky. The star is the point of interest, the reference plane is the local area around an observer on Earth's surface, and the reference vector points to true north. The azimuth is the angle between the north vector and the star's vector on the horizontal plane.
  Azimuth is usually measured in degrees. The concept is used in navigation, astronomy, engineering, mapping, mining, and ballistics.
context_section_description: 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 α is given by
  A better approximation assumes the Earth is a slightly-squashed sphere (an oblate 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 immeasurably 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., GeoScience Australia site. 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 f is the flattening, and e the eccentricity, for the chosen spheroid (e.g., ​¹⁄298.257₂₂₃₅₆₃ for WGS84) then
  If φ₁ = 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 φ₂ with declination and longitude difference with hour angle, and change the sign (since the hour angle is positive westward instead of east).
caption: The azimuth between Cape Town and Melbourne along the geodesic (the shortest route) changes from 141° to 42°. Azimuthal orthographic projection and Miller cylindrical projection.
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