Sight reduction

Sight reduction is the process of deriving from a sight the information needed for establishing a line of position.

Sight is defined as the observation of the altitude, and sometimes also the azimuth, of a celestial body for a line of position; or the data obtained by such observation.[1]

Nowadays sight reduction uses the equation of the circle of equal altitude to calculate the altitude of the celestial body,

and the azimuth Zn is obtained from Z by:

With the observed altitude Ho, Hc and Zn are the parameters of the Marcq St Hilaire intercept for the line of position:

Correction to the sextant altitude
Marcq St Hilaire intercept for the line of position

With B the latitude (+N/S), L the longitude (+E/-W), LHA = GHA + L is the local hour angle, Dec and GHA are the declination and Greenwich hour angle of the star observed, and Hc the calculated altitude. Z is the calculated azimuth of the body.

Basic procedures involved computer sight reduction or longhand tabular methods.

Tabular sight reduction

The methods included are:

Longhand haversine sight reduction

This method is a practical procedure to reduce celestial sights with the needed accuracy, without using electronic tools such as calculator or a computer. And it could serve as a backup in case of malfunction of the positioning system aboard.


The first approach of a compact and concise method was published by R. Doniol in 1955[2] The altitude is derived from sin(Hc) = na (m + n), in which n = cos(BDec), m = cos(B + Dec), a = hav(LHA).

The calculation is:

n = cos(B - Dec)
m = cos(B + Dec)
a = hav(LHA)
sin_Hc = n - a (m + n)
Hc = arcsin(sin_Hc)

Ultra compact sight reduction

Haversine Sight Reduction algorithm

A practical and friendly method using haversines was developed between 2014 and 2015,[3] and published in NavList.

A compact expression for the altitude was derived[4] using haversines, hav, for all the terms of the equation:

hav(ZD) = hav(B - Dec) + (1 - hav(B - Dec) − hav(B + Dec)) hav(LHA)

where ZD is the zenith distance

Hc = (90 - ZD) the calculated altitude

The algorithm if absolute values are used is:

if same name for latitude and declination
 n = hav(|B| - |Dec|)
 m = hav(|B| + |Dec|)
if contrary name
 n = hav(|B| + |Dec|)
 m = hav(|B| - |Dec|)
q = n + m
a = hav(LHA)
hav(ZD) = n + (1 - q) a
ZD = invhav -> look at the haversine tables
Hc = 90° - ZD

For the azimuth a diagram[5] was developed for a faster solution without calculation, and with an accuracy of 1°.

Azimuth diagram by Hanno Ix

This diagram could be used also for star identification.[6]

An ambiguity in the value of azimuth may arise since in the diagram 0 ≤ Z ≤ 90°. Z is E/W as the name of the meridian angle, but the N/S name is not determined. In most situations azimuth ambiguities are resolved simply by observation.

When there are reasons for doubt or for the purpose of checking the following formula[7] should be used.

hav(Z) = [hav(90° - Dec) - hav(B - Hc)] / (1 - hav(B - Hc) - hav(B + Hc))

The algorithm if absolute values are used is:

if same name
 a = hav(90° - |Dec|)
if contrary name
 a = hav(90° + |Dec|)
m = hav(B + Hc)
n = hav(B - Hc)
q = n + m
hav(Z) = (a - n) / (1 - q)
Z = invhav -> look at the haversine tables
if Latitude N:
 if LHA > 180°, Zn = Z
 if LHA < 180°, Zn = 360° − Z
if Latitude S:
 if LHA > 180°, Zn = 180° − Z
 if LHA < 180°, Zn = 180° + Z

This computation of the altitude and the azimuth needs a haversine table. For a precision of 1 minute of arc, a four figure table is enough.[8]

An example

 B = 34° 10.0′ N (+)
 Dec = 21° 11.0′ S (-)
 LHA = 302° 43.0′
Altitude Hc:
 a = 0.2298
 m = 0.0128
 n = 0.2157
 hav(ZD) = 0.3930 -> table ->
 ZD = 77° 39′
 Hc  = 12° 21′
Azimuth Zn:
 a = 0.6807
 m = 0.1560
 n = 0.0358
 hav(Z) = 0.7979
 Zn  = 126.6°

See also


  1. The American Practical Navigator (2002)
  2. . Table de point miniature (Hauteur et azimut), by R. Doniol, Navigation IFN Vol. III Nº 10, Avril 1955 Paper
  3. Rudzinski, Greg (July 2015). Ix, Hanno. "Ultra compact sight reduction". Ocean Navigator. Portland, ME, USA: Navigator Publishing LLC (227): 42–43. ISSN 0886-0149. Retrieved 2015-11-07.
  4. Altitude haversine formula by Hanno Ix
  5. Azimuth diagram by Hanno Ix.
  6. Hc by Azimuth Diagram
  7. Azimuth haversine formula by Lars Bergman

External links

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