# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**Re: An All-Haversine Azimuth (from Lat, Dec and dLon alone)?**

**From:**Willi Strohl

**Date:**2021 Mar 16, 02:27 -0700

Hello Tony,

If you are interested in a method of finding a position line only by using haversine or archaversine functions, here is an alternative way to do so:

The idea is to use two arbitrarily chosen latitudes (in the vicinity of your assumed position) and calculate the corresponding longitudes. So you get two geographically defined points. – The line, which goes through both points is the desired position line. - No calculation of the azimuth is necessary.

**Precedure:**

**1.** Select your celestial body, measure the height **h** with the sextant and record the time **UT1** of the measurement (as usual).

**2.** Estimate the appropriate **Grt** and the declination **δ** of the selected celestial body using the Nautical Almanac (as usual).

**3.** Pick two latitudes ϕ_{1} and ϕ_{2 }which are** roughly** in the vicinity of your currently assumed position. They should be about 5 to 20 nm apart from each other, depending on the scale of the nautical chart you are using.

**4.** Calculate the longitude ∆λ_{1} which corresponds to the latitude ϕ_{1 } by using the haversine formula:

**hav(∆λ _{1})**

_{ }**=**

**[**

**hav (**

**90-h**

**) - hav (**

**ϕ**

_{1}**-**

**δ**

**)**

**] / [1 –**

**hav (**

**ϕ**

_{1}**+**

**δ**

**)**

**-**

**hav (**

**ϕ**

_{1}**-**

**δ**

**)**

**]**

∆λ_{1} = archav[hav(∆λ_{1})] , note that there are always **2 possible solutions**: ∆λ can be positive or negative!

**5.** Calculate the corresponding preliminary longitudes λ_{11} and λ_{12}:

λ_{11}* = + ∆λ_{1} – **Grt ****à**** **If |λ_{11}*| < 180° then **λ**_{11}**= ****λ**_{11}******* ; **If λ_{11}* < -180° then **λ**_{11}**= ****λ**_{11}*** + 360°**

λ_{12}* = - ∆λ_{1} – **Grt ****à**** **If |λ_{12}*| < 180° then **λ**_{12}**= ****λ**_{12}******* ; **If λ_{12}* < -180° then **λ**_{12}**= ****λ**_{12}*** + 360°**

**6. Of λ**_{11}** and ****λ**_{12}** pick the one which is closest to your assumed position. This is the corresponding longitude ****λ**_{1 }**to the chosen latitude ϕ**_{1.}

**7.** Repeat steps 4. to 6. in order to get λ_{2} using ϕ_{2} in the haversine formula (step4.).

As final result you will get the two desired coordinate pairs **ϕ**_{1}**/**** λ**** _{1}** and

**ϕ**

_{2}**/**

**λ**

_{2}**to draw your line of position.**

**Important for the calculation:**

Latitudes ϕ and declinations δ can take values from -90° to +90°,

**(-)** corresponds to **southern** latitudes or declinations

**(+)** corresponds to **northern** latitudes or declinations

Longitudes can take values from -180° to +180°

**(-)** corresponds to longitudes **west** of Greenwich

**(+)** corresponds to longitudes **east** of Greenwich

**Advantages of this method:**

+ No exact dead reckoning position is required, you just need to know roughly where you are.

+ Only one haversine table and the nautical almanac are needed to solve the problem

**Disadvantages:**

- You need to do a **manual division** in step 4. for each coordinate pair (or use a calculator).

- The procedure doesn’t work, if the celestial body has an azimuth of 180° or 0° (or close to it). If you chose Polaris for example, there will only be one possible latitude. The line of position in this case would be the latitude itself.

Lycka till and best wishes

Willi