Sight reduction

The first approach of a compact and concise method was published by R. Doniol in 1955[4] and involved haversines. The altitude is derived from ${\displaystyle \sin(Hc)=n-a\cdot (m+n)}$, in which ${\displaystyle n=\cos(Lat-Dec)}$, ${\displaystyle m=\cos(Lat+Dec)}$, ${\displaystyle a=\operatorname {hav} (LHA)}$.

The calculation is:

n = cos(Lat − Dec)
m = cos(Lat + Dec)
a = hav(LHA)
Hc = arcsin(n − a ⋅ (m + n))

Haversine Sight Reduction algorithm

A practical and friendly method using only haversines was developed between 2014 and 2015,[5] and published in NavList.

A compact expression for the altitude was derived[6] using haversines, ${\displaystyle \operatorname {hav} ()}$, for all the terms of the equation: ${\displaystyle \operatorname {hav} (ZD)=\operatorname {hav} (Lat-Dec)+\left(1-\operatorname {hav} (Lat-Dec)-\operatorname {hav} (Lat+Dec)\right)\cdot \operatorname {hav} (LHA)}$

where ${\displaystyle ZD}$ is the zenith distance,

${\displaystyle Hc=(90^{\circ }-ZD)}$ is the calculated altitude.

The algorithm if absolute values are used is:

if same name for latitude and declination (both are North or South)
 n = hav(|Lat| − |Dec|)
 m = hav(|Lat| + |Dec|)
if contrary name (one is North the other is South)
 n = hav(|Lat| + |Dec|)
 m = hav(|Lat| − |Dec|)
q = n + m
a = hav(LHA)
hav(ZD) = n + a · (1 − q)
ZD = archav() -> inverse look-up at the haversine tables
Hc = 90° − ZD

For the azimuth a diagram[7] 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.[8]

An ambiguity in the value of azimuth may arise since in the diagram ${\displaystyle 0^{\circ }\leqslant Z\leqslant 90^{\circ }}$. ${\displaystyle Z}$ 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[9] should be used:

${\displaystyle \operatorname {hav} (Z)={\frac {\operatorname {hav} (90^{\circ }\pm \vert Dec\vert )-\operatorname {hav} (\vert Lat\vert -Hc)}{1-\operatorname {hav} (\vert Lat\vert -Hc)-\operatorname {hav} (\vert Lat\vert +Hc)}}}$

The algorithm if absolute values are used is:

if same name for latitude and declination (both are North or South)
 a = hav(90° − |Dec|)
if contrary name (one is North the other is South)
 a = hav(90° + |Dec|)
m = hav(|Lat| + Hc)
n = hav(|Lat| − Hc)
q = n + m
hav(Z) = (a − n) / (1 − q)
Z = archav() -> inverse look-up 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.[10][11]

Data:
 Lat = 34° 10.0′ N (+)
 Dec = 21° 11.0′ S (−)
 LHA = 57° 17.0′
Altitude Hc:
 a = 0.2298
 m = 0.0128
 n = 0.2157
 hav(ZD) = 0.3930
 ZD = archav(0.3930) = 77° 39′
 Hc = 90° - 77° 39′ = 12° 21′
Azimuth Zn:
 a = 0.6807
 m = 0.1560
 n = 0.0358
 hav(Z) = 0.7979
 Z = archav(0.7979) = 126.6°
 Because LHA < 180° and Latitude is North: Zn = 360° - Z = 233.4°