241. THE MERCATOR PROJECTION

An idea of how the Mercator projection is constructed will be obtained from fig. 62-1. The spherical surface of the Earth is expanded into a cylinder. A particular Mercator Chart may then be imagined as a small section “peeled” from the cylinder and unrolled flat. The graticule on this chart, however, is not a true or perspective, projection of the meridians and parallels, formed by drawing straight lines from the centre of the sphere through certain points on the surface of the sphere and noting corresponding points on the circumscribing cylinder. The Mercator chart is mathematically constructed. This means that the distance representing the spacing of meridians and parallels of latitude must be calculated; it is not possible to construct an accurate Mercator chart geometrically.

A German cartographer Gerhard Kaufman first used the principle on which the Mercator chart is constructed in 1569. Kaufman was known by various names, changing to Gerhard Krehmer when he settled in Belgium but eventually adopting the name Mercator (this being the Latinised version of his original name, which in English means “merchant”. Mercator’s ‘Map of the World’ was not an immediate success, and Mercator died taking the secret of his projection with him. By the turn of the century, however, the English mathematician, Edward Wright, solved the mystery of Mercator’s projection and in fact found certain errors in Mercator’s graticule, publishing a description and also a table for facilitating the construction of a Mercator chart. Charts drawn on the Mercator projection became increasingly popular as their worth became apparent, and by the 1630’s were in general use at sea.

The reason the Mercator projection became so popular for marine use was because it gives the chart the properties necessary for the navigator, namely that:

Rhumb-lines on the Earth appear as straight lines, and Angles between these rhumb-lines on the Earth are unaltered as they appear on the chart, so that the Equator (which is a rhumb-line as well as a Great-Circle) appears on the chart as a straight line, parallels of latitude appear as straight lines parallel to the Equator, and meridians appear as straight lines perpendicular to the Equator.

242.  THE GNOMONIC PROJECTION

The gnomonic projection is a geometrical projection where the sphere’s surface is projected outwards from the sphere’s centre onto a plane which is tangential to the sphere. The tangential point is usually at the centre of the area which is to be represented, and if the tangential point is at one of the Earth’s poles the projection is known as a polar gnomonic. Fig. 62-2 shows the graticule of a polar gnomonic chart produced by projecting the parallels and meridians (on the sphere) on to a tangent plane. The tangent plane touches the sphere at the Pole and is shown after being rotated 90° to a vertical position. The point of projection is at the centre of the sphere, O. Imagine the parallels and meridians drawn on a transparent sphere with a light at O; the graticule would appear on a flat sheet of paper as shown in Fig. 62-2. All meridians appear as straight lines which intersect at the point which represents the Pole, and any Great Circle appears as a straight line on this projection (see Fig. in the chapter on ‘The Mathematics of Navigation’).

A gnomonic chart which has the tangential point at any position other than the Pole is very difficult to construct. In such a case, the parallels of latitude appear as hyperbolic curves. Harbour plans and charts with a natural scale greater than 1/50,000 are drawn on the gnomonic projection because, the area to be represented being small, the tangential plane almost coincides with the surface of the Earth and the lack of parallelism of the meridians and parallels of latitude is so small as to be hardly detectable because of the smallness of the area which it represents.

Gnonomic charts are essential for Polar navigation because of the distortion of the Mercator projection in these high latitudes renders Mercator charts useless, while Gnonomic charts of the major oceans of the world are used for plotting Great Circle tracks for transference to Mercator charts for practical navigation.

243. SCALE ON THE MERCATOR CHART

Since the Equator is shown on the Mercator chart as a straight line of definite length, then the longitude scale is fixed by that length and must be constant in all latitudes because the meridians appear as straight lines perpendicular to the Equator. We know from the Parallel Sailing formulae (No.2 on page N14/2) that Departure = d.long.. x cos.lat. If the scale on a particular chart is ‘y’ inches to 1’ of longitude, then

DEPARTURE (Y INCHES) = 1’ CX COS.LAT., AND SINCE D.LONG. = DEP. X SEC.LAT., 1’ = Y X SEC.LAT.

In other words, one mile on the chart in a particular latitude is represented by y.sec.lat.inches, from which it follows that the scale of latitude and distance at a certain place on the chart is proportional to the secant of the latitude of that place.

Similarly, the amount of distortion on a Mercator chart is governed by the secant of the latitude of the longitude. Hence Greenland in Lat. 70º N. appears as a broad as Africa at the Equator, although in fact, Africa is three times as broad as Greenland (sec.70º =3).

Mercator charts are graduated along the top and bottom edge for longitude and on the left and right-hand edges for latitude and distance. The longitude scale should be used only for laying down or reading-off the longitude of a place, never for measuring distance. To measure distance on a Mercator chart, the length of a rhumb-line between two places is taken as the distance between them, but must be measured as shown in Fig. 62-3.

ZABCY is the rhumb-line as it appears on the chart. XX’, AA’, BB’ . . .  etc. are parallels of latitude. The distance XA must be measured on the latitude scale between X’ and A’, the distance AB on the scale between A’ and B’ and so on. If XY is not large, say less than 100 miles, no appreciable error results by measuring it on the latitude scale roughly on either side of its (XY’s) middle point.

244. MERIDIONAL PARTS

Since the latitude and distance scale at any place on a Mercator chart is proportional to the secant of the latitude at that place, the scale continually increases as it recedes from the Equator until, at the Pole, it becomes infinite (it is for this reason that the complete polar regions cannot be shown on Mercator charts, and Gnonomic charts are used for these areas). The latitude scale thus affords no ready means of comparison with the fixed longitude scale. 

In Fig. 62-4. the tangent of the course ∠ZXY is not YZ (measured on the longitude scale) divided by XZ (measured on the latitude scale). 

For the ratio  YZ

                          XZ   to be valid, YZ and XZ must be measured in the same fixed units, and the most convenient unit to use is one minute of arc on the longitude scale, called a “Meridional part”. The number of these longitude units contained in the length of a meridian from the Equator to the parallel of any latitude is known as the “Meridional parts” for that latitude.

The Meridional parts for any latitude are tabulated in all Nautical tables, and since the Earth is a spheroid, then the “Meridional Parts for the Terrestrial Spheroid” should be used in preference to those for a true sphere, although in the ordinary practice of Navigation, any error arising from using Meridional parts for the sphere instead of for the spheroid would be negligible.

To find the Meridional parts for Lat.45, for example, enter any nautical tables at the table of “Meridional Parts for the Spheroid”, where opposite 45 will be found 3013.38 mer.parts.

If the longitude scale of a particular Mercator chart is such that one inch represents one degree (60 mer-parts), then the length of the meridian from the Equator to the parallel of 45º is  3013.38

                                                                                 60       = 50.2 inches. 

From this it will be seen that Meridional parts involve chart lengths, are not in any way connected with distance on the Earth’s surface, which is measured in nautical miles. Meridional parts are used in two applications; (i) the construction of Mercator charts, and (ii) Mercator Sailing.

245. DIFFERENCE OF MERIDIONAL PARTS (D.M.P.)

We said above that for the tangent of the rhumb-line course ∠ZXY (Fig. 62-4) to equal  YZ,

                                                                                                                                                                     XZ 

the lengths YZ and XZ must be measured in the same fixed units, and we have now established that these units are Meridional parts (or 1’ of longitude). The length YZ is simply the d.long. Between the two places at the ends of the rhumb-line, X and Y. The length XZ is the difference of the Meridional parts for the latitude of X and Y respectively, or the number of Meridional parts in the length of the meridian between the parallels through X and Y. The difference in Meridional parts is always written “d.m.p.”, and is found simply by taking the differences between the values of Meridional parts extracted from the tables for the respective latitudes X and Y, subtracting the two values if X and Y are on the same side of the Equator, and adding them if X and Y are on opposite sides of the Equator. This will become clear by studying Fig. 62-4.

246. MERCATOR SAILING

Mercator Sailing is the most modern of the Rhumb-Line Sailings and is derived from the representation of the Plain Sailing triangle on the Mercator chart. Any Plane Sailing triangle would be represented on a Mercator chart by a similar but larger triangle (because the meridians are parallel on the chart and do not converge as on the sphere).

In Fig. 62-5, triangle XYZ is a Plane Sailing triangle and triangle X’Y’Z’ is its representation on a Mercator chart. In the Plane Sailing triangle, all three sides, representing dep., d.lat., and distance are on the same scale, i.e., the latitude scale or nautical miles. In the chart triangle, the scale used is the longitude (or Meridional parts) scale and the side opposite to the course angle is labelled d.long. This is because the longitude scale on a Mercator chart is constant for all latitudes.

The adjacent side X’Y’, however, on the same scale (i.e., in Meridional parts) represents the d.m.p. between the latitudes of X and Y (or X’ and Y’). Note that the hypotenuse X’Y’ is not the true distance, which can only be measured in nautical miles.

When the two triangles are superimposed, as in Fig. 62-5, it will be seen that the course angle (∠ZXY or ∠Z’X’Y’) is the same for both triangles, so that tan. Course (ΔX) =    dep.  =  d.long.  from which the four 

Mercator Sailing formulae are derived: –

                                                                     d.lat. d.m.p.

Tan.Course =   d.long.                             ………. (1)

                            d.m.p.

Dist. = d.lat. x sec.Course                     ………. (2)

d.lat. = dist x cos.Course                      ………. (3)

d.long. = d.m.p. x Tan.Course             ………. (4)

Of these formulae, only (1) and (4) are new, as (2) and (3) occur in Plane Sailing.

Strictly speaking, a fifth formulae should be added to these Mercator Sailing formulae. If the table of natural secants is inspected, it will be seen that for a small change in the value of a large angle, the secant changes considerably.

Thus if the course angle is great (i.e., more than about 60º), error in the calculated distance may be introduced because of the difficulty in interpolating an accurate value of the secant of the course angle. Consequently, instead of using formulae (2) above: dist. = d.lat. x sec.Course, it would be more accurate in cases where the course angle is large to substitute: dist. = d.lat. x tan.Course x Cosec.Course . . . . . . . . (2A).

In the table of natural cosecants it will be seen that for a small change in the value of a large angle, the cosecant does not change appreciably.

As with the other Rhumb-Line Sailings, the use of Mercator Sailing is in solving the two most common navigational problems, namely, finding the course and distance along a rhumb-line track, and finding the position after sailing a known course and distance. Both of these problems can be solved equally well by either Middle Latitude or Mercator Sailing, but the Mercator method eliminates the labour of finding the true middle latitude and is therefore preferable.

The procedure followed in Mercator Sailing is as follows: –

To Find the Course and Distance: 

Calculate the d.lat. and d.long.

Using “Meridional Parts for the Spheroid” tables, calculate the d.m.p.

Using Mercator Sailing formula (1), calculate the course angle.

 Using Mercator Sailing formulae (2) (or (2A) for angles over 60º), calculate the distance.

To Find the Final Position:  

Using Mercator Sailing formulae (3), calculate the d.lat.

Apply the d.lat. to the initial lat, to find the final lat.

Using “Mer.Parts for the Spheroid” tables, calculate the d.long.

Apply the d.long. to the initial long. To find the final long.

The following examples will illustrate the use of Mercator Sailing: –

From Monaco to Elba, Course = 113¼º T., Distance = 127.2 miles

(which compare with result by Traverse Table on page N14/8)