Laplace Azimuth

the geodetic azimuth A of the direction to an observed point obtained from its astronomical azimuth a corrected by taking into account the effect of the deviation of the vertical at the place of observation. The astronomical azimuth of a direction to a point in space is the dihedral angle between the plane of the astronomical meridian of the place of observation and the plane that passes through the vertical at this place and the observed point. The Laplace azimuth (geodetic azimuth) of a point in space is equal to the dihedral angle between the plane of the geodetic meridian of the place of observation and the plane that passes through the normal to the surface of the reference ellipsoid at this place and the observed point. The following formula is used to change from the astronomical azimuth to the Laplace azimuth:

A = α − η tan Φ − (ξ sin α − η cos α) cot z

in which ξ and η are the components of the deviation of the vertical at the place of observation in the planes of the meridian and the prime vertical, Φ is the latitude of the place of observation, and z is the zenith distance of the observed point in space. For z close to 90°, this formula leads to the Laplace equation for determining the Laplace azimuth: a − A = η tan Φ (the equation was named after P. Laplace, who established the relationship).