## 8.36 Rhumb Line Distance and Course Computer

According to the Merriam-Webster dictionary, the definition of the word “rhumb” is (a) a line or course on a single bearing, (b) any of the points of the mariner’s compass. That is, “rhumb” signifies N, NNE, NE, ENE, E, etc. But the term can actually refer to any angle of direction or *bearing* relative to due North. In navigation by air or sea, bearing angle is measured with respect to a Meridian – an imaginary Great Circle that passes through the vessel’s location and through both the North and South poles of the earth. The angle is typically measured in the easterly direction from due North. So a vessel with a bearing angle of 45 degrees would be heading NE, but if it turned around to head SW its bearing would be 225 degrees.

A course with constant bearing means that the vessel will cross every Meridian at that bearing angle. With a compass needle that is constantly pointing due North, say, then the pilot can stay on “course” by maintaining a constant bearing, or “rhumb”, relative to the needle’s direction.

The shortest path between two locations on a globe is actually along a Great Circle. Along a Great Circle, however, the bearing of a craft will change along the way, as this trajectory will cross Meridian lines at differing angles, unless the Great Circle is exactly along the equator or if the vessel is traveling exactly North or South. It is possible though to find a trajectory between two locations for which the bearing does not change. A general trajectory for which its course or bearing is kept constant is called a “rhumb line”. The figure below illustrates the difference between a course along a Great Circle and a course along a Rhumb Line. While not as short a distance, traveling along the rhumb line between two points implies that a bearing is to be set once and then maintained, thus simplifying the piloting of the vessel in principle.

The “Rhumb Line Distance and Course Computer” shown in the image below was created by John E. Clemons of San Antonio, TX, and John G. Nelson of Houston, TX. They applied for a U.S. patent of their device on April 6, 1944, during the Second World War and the patent, No. 2,405,113, was granted two years later in August of 1946. As the words “Patent Pending” are on the instrument, it was likely made between these two dates. It was made by Simplified Flight Calculator Co., San Antonio, TX.

### 8.36.1 The Rhumb Course Angle

Even though we live on a sphere, if the two end points of a trajectory are close enough together the path can be treated as a trajectory between two points on a plane. If we know that the E-W and N-S coordinates of our two nearby points are \((x_1,y_1)\) and \((x_2,y_2)\), then the angle of our course angle \(\alpha\) from point 1 to point 2 would be given by \(\tan\alpha = \Delta x/\Delta y,\) remembering that the angle is taken relative to due North. We would find the distance between our two points using \(\Delta s = \sqrt{\Delta x^2 + \Delta y^2}.\) But when sailing across an ocean, or flying the same course at relatively low altitudes, the curvature of the earth must be taken into account.

Using common coordinates of longitude (E-W) and latitude (N-S), we can express distances and angles using spherical coordinates. Let the latitude, noted as \(L_a\), be the angle from the Earth’s equator toward a pole, and let the azimuthal angle or longitude, noted as \(L_o\), be measured Eastward from a specified Meridian. Paths of constant latitude are circles with radii that vary according to \(r_a = R_e \cos L_a\), where \(R_e\) is the radius of the earth (here, taken to be a perfect sphere). Hence, if one travels between two nearby points given by \((L_o, L_a)_1\) and \((L_o, L_a)_2\), then the local course angle would be defined by

\[ \tan\alpha = \frac{\Delta x}{\Delta y} = \frac{R_e \cos L_a\; \Delta L_o}{R_e\;\Delta L_a} = \cos L_a\;\frac{\Delta L_o}{ \Delta L_a}. \]

If Points 1 and 2 become further and further apart, then during the journey between them the local latitude \(L_a\) can change. In this case, to maintain a constant course angle, then the rate of change of \(L_o\) with respect to \(L_a\) must change as well. So suppose we want to maintain a constant \(\alpha\). Then from our equation above we need to have

\[ \Delta L_o = \tan\alpha\;\frac{ \Delta L_a}{\cos L_a} = \tan \alpha \cdot \sec L_a \;\Delta L_a \]

which, for continuous changes in latitude and longitude, we then can use *calculus* to sum up the effect and get

\[ \int_{{L_o}_1}^{{L_o}_2} dL_o = \tan\alpha \cdot \int_{{L_a}_1}^{{L_a}_2} \sec L_a \;dL_a = \tan\alpha \cdot \ln | \tan L_a + \sec L_a ~|_{{L_a}_1}^{{L_a}_2}~ \\ ~\\ {L_o}_2 - {L_o}_1 = \tan\alpha \cdot \left\{~ \ln | \tan {L_a}_2 + \sec {L_a}_2 | - \ln | \tan {L_a}_1 + \sec {L_a}_1 |~\right\}. \]

In other words, if we define the function

\[ \Sigma(u) \equiv \ln | \tan u + \sec u | ~~~~ = ~~~~ \ln | \tan ( u/2 + π/4 ) | \]

(using standard trig identities), then along the rhumb line between points 1 and 2 the constant course angle is given by

\[ \tan\alpha =\frac{L_{o2} - L_{o1} }{ \Sigma(L_{a2}) -\Sigma(L_{a1}) }. \]

A plot of \(\Sigma(L_a)\) is shown below.

```
Sig = function(x){
z=x/180*pi # convert to radians, than back to degrees
log( (sin(z) + 1)/cos(z) )*180/pi
}
```

We see that \(\Sigma\) skyrockets near 90 degree latitudes, with \(\Sigma\rightarrow\infty\) at 90 degrees, but remains fairly well behaved up to about 85 degrees or so. The blue line is for \(\tilde \Sigma(L_a) = L_a\), showing the “plane-like” behavior near the equator.

As a numerical example, let’s take \((L_o, L_a)_1\) = (15, 30) degrees and \((L_o, L_a)_2\) = (43, 45) degrees. We compute the course:

For this example we get \(\alpha\) = 55.8 degrees for the desired bearing or “course”.

### 8.36.2 Rhumb Line Distance

To find the distance traveled along a rhumb line between two points, we note that an infinitesimal path length along any curve on the surface of the (spherical) earth is given by

\[ ds = \sqrt{dx^2 + dy^2} \longrightarrow \sqrt{(R_e\cos L_a dL_o)^2 + (R_edL_a)^2}. \]

If the path is exactly along a constant value of \(L_a\), then \(dL_a=0\) giving \(ds = R_e\cos L_a\; dL_o\). (Note that for these equations, we need \(dL_a\) and \(dL_o\) in radians, not degrees.) But if the latitude is changing along the path, then

\[ ds = R_e \sqrt{1+(\cos L_a \;dL_o/dL_a)^2 }~~dL_a = R_e \sqrt{1+\tan^2\alpha }~~dL_a. \]

So, after having computed our course \(\alpha\) to send us from Point 1 to Point 2, we find that the total distance that will be traveled is

\[ s = R_e\sqrt{1+\tan^2\alpha}\cdot |L_{o2}-L_{o1}|. \]

For our example above, we get a value of \(s\) = 1846 miles = 1604 nautical miles. (The conversion is 1.15078 miles per nautical mile.)

### 8.36.3 Mercator Projection

If we go back for a minute to our definition of \(\Sigma\) as a function of latitude, we had

\[ \Sigma(L_a) \equiv \int_0^{L_a} \sec L \; dL. \]

We previously noted that along a constant latitude \(L_a\), the distances between different meridians (\(\Delta L_o\)’s) shrink by a factor of \(\cos L_a\). If we took the azimuthal dimension along each latitude circle and *stretched* it by a factor of \(1/\cos L_a = \sec L_a\), then the sphere would be mapped into a cylinder. And cutting the cylinder along a line parallel to its axis, and then unfolding it, one would get a two-dimensional map. Finally, by using the factor used for creating the spacing of latitude lines – our function \(\Sigma(L_a)\) – to stretch the vertical scale, we arrive at a square map that preserves local angles. (This is called a conformal map.) This particular conformal mapping is called a Mercator projection, first proposed by geographer and mapmaker Gerardus Mercator in 1569. An example is the world map shown below, with its familiar feature of having apparently exaggerated sizes of land masses near the poles (Greenland, Antarctica, etc.).

On such a Mercator map, the horizontal and vertical distances (\(x\) and \(y\), say) measured on the map are given by

\[ x = R_e L_o ~, ~~~~~~~~~~~~ y = R_e \cdot \Sigma(L_a) = R_e\ln | \tan ( L_a/2 + π/4 ) | \]

though it must be pointed out that Mercator created his early maps prior to the discovery of logarithms. He evidently (though this is not documented) *summed* successive values of \(\sec L_a\cdot \Delta L_a\) to estimate values of what we have called \(\Sigma(L_a)\).

So why was this type of scaling chosen? With this specific projection, if one draws a line on the map between two points, the line will have a slope \(m\) relative to each local Meridian (i.e., a slope relative to due North) given by

\[ {\rm slope} = \tan\alpha = \frac{\Delta x}{\Delta y} ~~~~ = ~~~~ \frac{\Delta L_o}{\Delta \Sigma(L_a)} \]

So on a Mercator map, one can draw a straight line between two points using a ruler, measure \(\Delta x\) and \(\Delta y\) on the map, and directly extract the rhumb course \(\alpha\) between the two points using a table of values of tangents, for example. Such a map was a significant advancement in navigation, allowing pilots to find a course between two locations in a straightforward manner. It was the use of the Mercator projection on navigational maps that led to the creation of the term “rhumb line” to describe a course of constant bearing across the globe.

#### 8.36.3.1 Connection to Gudermann

Now suppose the distances \(x\) and \(y\) are measured from the origin on a Mercator map. Since

\[ x = R_e L_o ~, ~~~~~ {\rm and} ~~~~~~~~y = R_e \cdot \Sigma(L_a) =R_e \ln | \tan(L_a/2+\pi/4) | , \]

then, the corresponding longitude is

\[ L_0 = x/R_e \] and solving for \(L_a\) we find the latitude

\[ L_a = 2\tan^{-1}(e^{y/R_e}) -\frac{\pi}{2}. \]

Interestingly, the function for \(L_a\) above is found elsewhere in the slide rule world. Often designated as \({\rm gd}(x)\), this is known as the Gudermannian function,

\[ {\rm gd}(x) = \int_0^{x} \frac{du}{\cosh u} = 2\tan^{-1}(e^{x}) -\frac{\pi}{2}. \]

A \({\rm gd}(x)\) scale is found on the Hemmi Model 153 slide rule, used for finding the values of hyperbolic trigonometric functions. A discussion of this function and its use on the 153 can be found in the vignette Hemmi 153 Scales.

So, if you measure the horizontal and vertical distances from the origin to a point (\(x,y)\) on the Mercator map, the transformations that give longitude and latitude are:

\[ L_o = \frac{x}{R_e} ~ , ~~~~~~~~~~~~ L_a = {\rm gd}\left(\frac{y}{R_e}\right) \]

or, going the other way around, to make a Mercator map from coordinates on the globe,

\[ x = R_e\cdot L_o ~, ~~~~~~~~~~~~ y = R_e \cdot gd^{-1}( L_a). \]

#### 8.36.3.2 Connection to Gunter

Rhumb line-related scales can also be found on a Gunter’s rule. One side of a Gunter rule from the Collection is shown in the image below.

The scale labeled `Num`

(line of numbers) is a 24-inch 2-decade log scale. We also see scales marked `Sin`

and `Tan`

which have lengths that go as the log of the sine or tangent of the angle indicated, as found on modern slide rules. The scale marked `V.S.`

refers to a *versed sine*. The modern definition of the versed sine is \(1-\cos\theta\). But in the the late 1500s it represented 1 minus the *haversine*, or \(\sin^2(\theta/2) =\frac12(1+\cos\theta).\)

As for the `S R`

and `T R`

scales, these stand for “Sine of Rhumbs” and “Tangent of Rhumbs”, respectively. They provide the logarithms of the sine and tangent of rhumb directions, where the points 1,2,…,8 correspond to N by E, NNE, NE by N, NE, NE by E, ENE, E by N, and E. That leaves the bottom two scales, `Mer`

and `E.P.`

. The last scale is a scale divided into “equal parts” and goes from right to left. And then, finally, `Mer`

stands for “Meridian”, or Meridian line on a Mercator map. For each latitude it provides the *excess* distance found between latitudes on the Mercator map. By comparing distances along `Mer`

with the `E.P.`

scale, one can ascertain the stretching of the Mercator coordinates. Below is our attempt using our function \(\Sigma(x)\) to reproduce the `Mer`

and `E.P.`

scales on the Gunter rule, which can be compared to the values (labeled `Gunter`

in the table below) of the scales shown in the above image.

EP | 90 | 80 | 70 | 60 | 50 | 40 | 30 | 20 | 10 |

Mer | 2169 | 130 | 89 | 65 | 48 | 34 | 21 | 10 | 0 |

Gunter | NA | 128 | 88 | 65 | 48 | 33 | 21 | 10 | 0 |

### 8.36.4 A Slide Rule Calculation

So, finally, we are in a position to discuss the Rhumb Line Distance and Course Computer. The Clemons and Nelson slide rule has two independent sides. The Front has scales labeled Latitude Difference (DL), Distance (DL), and course (COURSE). The Front has a single hairline rotatable cursor.

The Back of the rule has scales labeled Distance (DISTANCE) in Nautical Miles, Latitude Difference (DL), and a special set of spiral curves denoted as Longitude Difference (DLo). The Back also has a single hair-line cursor, but to use the spiral curves there is a special cursor-on-cursor, or “tab”, that slides along the main cursor. Over the length of the main cursor line is a scale labeled Mid-Latitude (ML). This is read using the tab and is to be used to take readings along the spiral curves, as will be explained in our example below. If the latitudes of the two end points are L1 and L2, then DL = L2-L1 while ML = (L1+L2)/2. In similar manner, DLo = Lo2-Lo1.

Clearly the course angle and the rhumb line distance depend not upon the difference of latitudes, but upon on the values of the latitudes themselves, due to the fact that the radii of constant latitudes are shrinking according to \(R_a = R_e\cos L_a\). So the final rhumb line course and distance must both be dependent upon the actual values of the initial and final latitudes. The Clemons-Nelson Computer uses the two parameters DL and ML to take this condition into account. Note that the curves plotted on the Computer are valid within the range of \(\pm 75\) degrees in latitude. Flying (or sailing) above or below these latitudes are typically not performed using rhumb lines, which in these regions would spiral around and around the poles.

So the basic calculations go something like this. Take two points, (Lo1,L1) and (Lo2,L2), then compute DL = L2-L1, DLo - Lo2-Lo1, and ML = (L1+L2)/2. On the back, place the arrow on the rotating disc over the value of DL on the DL scale. Place the sliding tab on the main cursor onto the value of ML. Rotate the main cursor until the the tab’s hairline and main cursor cross the spiral curve for the value of DLo. Under the hairline of the main cursor will be the Rhumb distance on the DISTANCE scale.

Next, flip the computer over to the Front. Here, rotate the cursor to place its hairline over DL on the DL scale. Rotate the disc until the DISTANCE found on the back is under the hairline. Look for the arrow on the disc labeled “INDEX” and above in, on the COURSE scale will be the bearing of the rhumb line \((\alpha\) in our notation). As is common for slide rule calculations, the quadrant of the angle will need to be thought through to determine a final course heading if negative values of DL and/or DLo are present.

We earlier computed the rhumb line distance and course for \((L_o, L_a)_1\) = (15, 30) degrees and \((L_o, L_a)_2\) = (43, 45) degrees. Re-doing the calculation using the slide rule, we use

- DL = \((L_a)_2 - (L_a)_1\) = 15\(^\circ\)

- DLo = \((L_o)_2 - (L_o)_1\) = 28\(^\circ\)

- ML = \(\frac12[(L_a)_1 + (L_a)_2]\) = 37.5\(^\circ\)

Setting the arrow on the Back to DL, and sliding the tab to ML, we rotate the main cursor until the ML crosshair intersects the curve for DLo as shown in the image below.

We see that under the main cursor’s hairline the DISTANCE scale shows 1605, which is consistent with our earlier answer.

Then, on the Front we move the cursor to DL and hold it, then rotate the disc until the DISTANCE we just found is under the hairline, as shown in our next image.

As we can see, the COURSE – found at the arrow labeled INDEX – is about 56 degrees, again as we found earlier.

### 8.36.5 Further Reading

- John E. Clemons and John G. Nelson, “Rhumb Line Calculator or Distance and Course Computer”, U.S. Patent 2,405,113 (1946). Also contains example calculations and further directions.

- P.V.H. Weems,
*Air Navigation*, Third Edition, McGraw-Hill, New York (1943).

- For a nice overview of the Mercator Projection, see https://en.wikipedia.org/wiki/Mercator_projection.

- Nicholas Rougeux,
*The Construction & Principal Uses of Mathematical Instruments*(2022), https://www.c82.net/math-instruments/, visited 21 Oct 2024.

- Otto van Poelje, “Gunter Rules in Navigation”,
*Jour. of Oughtred Soc.***13**.1 p11 (2004).

From P.V.H. Weems,

*Air Navigation*, Third Edition, McGraw-Hill, New York (1943).↩︎