Azimuth

Normal-section azimuth can be calculated as follows:^[citation needed]

${\displaystyle {\begin{aligned}e^{2}&=f(2-f)\\1-e^{2}&=(1-f)^{2}\\\Lambda &=\left(1-e^{2}\right){\frac {\tan \varphi _{2}}{\tan \varphi _{1}}}+e^{2}{\sqrt {\frac {1+\left(1-e^{2}\right)\left(\tan \varphi _{2}\right)^{2}}{1+\left(1-e^{2}\right)\left(\tan \varphi _{1}\right)^{2}}}}\\\tan \alpha &={\frac {\sin L}{(\Lambda -\cos L)\sin \varphi _{1}}}\end{aligned}}}$

where f is the flattening and e the eccentricity for the chosen spheroid (e.g., 1⁄298.257223563 for WGS84). If φ₁ = 0 then

${\displaystyle \tan \alpha ={\frac {\sin L}{\left(1-e^{2}\right)\tan \varphi _{2}}}}$

To calculate the azimuth of the Sun or a star given its declination and hour angle at a specific location, 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).^{[citation needed]}

The cartographical azimuth or grid azimuth (in decimal degrees) can be calculated when the coordinates of 2 points are known in a flat plane (cartographical coordinates):

${\displaystyle \alpha ={\frac {180}{\pi }}\operatorname {atan2} (X_{2}-X_{1},Y_{2}-Y_{1})}$

Remark that the reference axes are swapped relative to the (counterclockwise) mathematical polar coordinate system and that the azimuth is clockwise relative to the north. This is the reason why the X and Y axis in the above formula are swapped. If the azimuth becomes negative, one can always add 360°.

The formula in radians would be slightly easier:

${\displaystyle \alpha =\operatorname {atan2} (X_{2}-X_{1},Y_{2}-Y_{1})}$

Note the swapped ${\displaystyle (x,y)}$ in contrast to the normal ${\displaystyle (y,x)}$ atan2 input order.

The opposite problem occurs when the coordinates (X₁, Y₁) of one point, the distance D, and the azimuth α to another point (X₂, Y₂) are known, one can calculate its coordinates:

${\displaystyle {\begin{aligned}X_{2}&=X_{1}+D\sin \alpha \\Y_{2}&=Y_{1}+D\cos \alpha \end{aligned}}}$

This is typically used in triangulation and azimuth identification (AzID), especially in radar applications.

There is a wide variety of azimuthal map projections. They all have the property that directions (the azimuths) from a central point are preserved. Some navigation systems use south as the reference plane. However, any direction can serve as the plane of reference, as long as it is clearly defined for everyone using that system.

Comparison of some azimuthal projections centred on 90° N at the same scale, ordered by projection altitude in Earth radii. (click for detail)

If, instead of measuring from and along the horizon, the angles are measured from and along the

In mathematics, the azimuth angle of a point in

For magnetic tape drives, azimuth refers to the angle between the tape head(s) and tape.

In sound localization experiments and literature, the azimuth refers to the angle the sound source makes compared to the imaginary straight line that is drawn from within the head through the area between the eyes.

An azimuth thruster in shipbuilding is a propeller that can be rotated horizontally.

Part of a series on
Astrodynamics
Orbital mechanics
Orbital elements Apsis Argument of periapsis Eccentricity Inclination Mean anomaly Orbital nodes Semi-major axis True anomaly
Types of two-body orbits by eccentricity Circular orbit Elliptic orbit Transfer orbit (Hohmann transfer orbit Bi-elliptic transfer orbit) Parabolic orbit Hyperbolic orbit Radial orbit Decaying orbit
Equations Dynamical friction Escape velocity Kepler's equation Kepler's laws of planetary motion Orbital period Orbital velocity Surface gravity Specific orbital energy Vis-viva equation
Celestial mechanics
Gravitational influences Barycenter Hill sphere Perturbations Sphere of influence
N-body orbits Lagrangian points (Halo orbits) Lissajous orbits Lyapunov orbits
Engineering and efficiency
Preflight engineering Mass ratio Payload fraction Propellant mass fraction Tsiolkovsky rocket equation
Efficiency measures Gravity assist Oberth effect
Propulsive maneuvers Orbital maneuver Orbit insertion
v t e

Further reading

• Rutstrum, Carl, The Wilderness Route Finder, University of Minnesota Press (2000),
  ISBN 0-8166-3661-3

External links

Look up azimuth in Wiktionary, the free dictionary.

• "Azimuth" . Encyclopædia Britannica (11th ed.). 1911.
• "Azimuth" . Collier's New Encyclopedia. 1921.