The Rhumb Line

If a ship were to steer a steady course, that is one on which her heading remains constant, her track would cut all meridians at the same angle, as the following diagram shows.  Such a line on the Earth’s surface is called a rhumb line.

When the rhumb line cuts all meridians at 90^o, it will coincide with either a parallel of latitude or with the Equator.  When the angle is 0^o, the rhumb line will be along a meridian of longitude.  A vessel’s course will always be a rhumb line; thus the course to be steered to travel from one place to another will refer to the angle between the rhumb line joining the places and any meridian.

 Calculating the distance between two points along a rhumb line. In the next diagram, A, B, C, D and Z are meridians of longitude;  the lines aB, bC, and cD are different parallels of latitude; and the line ABCDZ is a rhumb line.  A series of right-angled triangles have been constructed along the rhumb line AZ and in each triangle, one short side lies along a meridian of longitude, one lies along a parallel of latitude and the hypotenuse lies along the rhumb line.

It can be seen from the diagram that the east-west distance between two points along the rhumb line is the sum of the distances along the parallels of latitude corresponding to the difference in longitude in each of the right-angled triangles.  This east-west distance is known as the departure

Middle Latitude. If we were to calculate the departure along each of the parallels of latitude aB, bC, cD, in the diagram above, we would find that they would not be equal and so the task of calculating the total departure would be complicated.  In practice, the total departure is taken to be the east-west distance along the intermediate of these parallels which is known as the ‘middle latitude’.

The formula to calculate departure is as follows:

Departure = d.long cos(middle latitude).

Mean Latitude.  In most cases, the arithmetic mean of the two latitudes can be used as the middle latitude without appreciable error, so the approximate formula dep.= d.long cos(mean lat) may be used.

When the difference of latitude is large (over 600 n.m.) or the latitudes are close to either of the poles, the middle latitude must be used instead of the mean latitude and in these cases, we have the more accurate formula:

Dep. = d.long cos(mid lat).

 The difficulty lies in the task of calculating the middle latitude which involves finding the mean of the secants all the intermediate latitudes by integration.  Such methods are obviously impracticable in situations where courses and distances have to be calculated rapidly at sea.  For this reason, tables of corrections to be applied to the mean latitude are contained in various collections of nautical tables.  Since, astro navigation involves short distance sailing calculations, it is not intended to copy middle latitude correction tables here; however, the following example demonstrates their use:

Suppose a ship sails from position 50^oN, 32^oE., to 70^oN., 15^oE.  The d.long is 17^o and the mean latitude is 60^o.

The formula for calculating departure using the mean lat. is:

dep.= d.long cos(mean lat)

Using this formula we have:

Dep. = 17^o cos(60)

= 1020’ cos(60)

= 510’ or 510 n.m.

In the tables for converting mean latitude to middle latitude, the correction for a mean latitude of 60^o and a difference of latitude of 20^o is +1^o 09’.  So the middle latitude = 61^o.15.

The formula for calculating departure using the middle latitude is:

Dep. = d.long cos(mid lat).

= 1020 cos (61.15)

= 492.17 n.m.

By comparing these results, we can see that there is a significant difference between calculations involving the mean latitude on one hand and the middle latitude on the other.

Summary of Formulae.  The formulae so far derived are summarised below:

Ddist  =  Dlong x Cos Lat.

Dlong  =   Ddist ÷ Cos Lat. = Ddist x Sec Lat.

dep.= d.long cos(mean lat) (for distances 600 n.m. or less).

Dep. = d.long cos(mid lat). (for distances over 600 n.m.).

 The Rhumb Line Formulae.

With the next diagram, we expand on the work above,

• The rhumb line AZ is divided into a large number of equal parts AB, BC, CD, DZ.
• aB, bC, cD… are arcs of parallels of latitude drawn through B, C, D…..
• Pa’, Pb’, Pc’… are meridians of longitude.
• Therefore, the angles at a, b, c….. are right angles.

If the divisions of AZ are made sufficiently small, the triangles ABa, BCb, CDc…..  will be small enough to be treated as plane triangles instead of spherical triangles.

• Since the course angle is constant by the definition of a rhumb line, these small triangles are equal.

Consider triangle ABa in the diagram;

AB is the distance made good, aB is the departure along a parallel of latitude, angle aAB is the course angle.

Therefore, Sin(course angle) = departure ÷ dist.  This formula applies to all of the small triangles since they are equal.

By transposition, the above formula becomes:

Dep = Dist x sin(course)

The departure between A and Z therefore, is the sum of the departures of all of the small triangles.

Therefore, by addition:

aB + bC + cD + …. = (AB + BC + CD + …. x sin(course)

i.e. Dep = Dist sin(course) 

If we again consider triangle ABa,  Aa = AB cos (course)

But Aa is the difference in Latitude (D.Lat) between A and B

So D.Lat = AB cos(course)

Again, this formula applies to all of the small triangles since they are equal.  Therefore, by addition, the total D.Lat corresponding to the total distance between A and Z becomes:

D.Lat = Dist cos(course)

We have established formulae to calculate Dep and D.Lat; we now need a formula to find the course.

If we return to triangle ABa, we can see that the course angle can be found by the formula:  tan(course) = Dep ÷ D.Lat.

As before, this formula applies to all of the small equal triangles.  So, by addition, the rhumb line course between A and Z can be found by the formula:

Tan(course) = Dep ÷ D.Lat  

Short Distance Sailing.

Short distance sailing is a term which is applied to sailing along a rhumb-line for distances less than 600 nautical miles.  From the formulae derived in this chapter, the following are used extensively in short distance sailing:

To Calculate Departure when the course is not known:

dep.= d.long cos(mean lat)

To Calculate Departure when the course is Known:

Dep  = Dist x Sin(course)

To Calculate Distance when departure and course are known:

Dist  =         Dep  ÷ Sin (course)

To Calculate Dlat when the distance and course are known:

DLat = Dist x Cos(course)

To Calculate Course to Steer(the rhumb line course between two points)

Tan(course) =   Dep  ÷ D.Lat

To calculate Dlong (difference in longitude corresponding to the departure):

DLong.  =  Dep. x Sec(Mean.Lat)   or Dlong =      Dep ÷ Cos(Mean.Lat)

Example.  At 0900 GMT, a life raft is reported to be in position 30^o 56’.4 S, 0^o 25’.6 E.  A rescue ship reports that its ETA at the vicinity is 2130 GMT.  The rescue ship’s navigator calculates that wind and ocean currents will cause the life raft to drift in direction 345^o at 3 knots.  Calculate the expected position of the life raft when the rescue ship is due to arrive.

Solution:

Click here for an exercise on  short distance sailing.

A fuller explanation of rhumb line sailing together with the derivation of the formulas given above can be found in the book ‘Astro Navigation Demystified’.

Link: The relationships between longitude and latitude and the nautical mile

web: www.astronavigationdemystified.com 

email: bookcase.learning@btinternet.com