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The Conformal Mercator Projection

Map projections are attempts to portray the surface of the earth or a portion of the earth on a flat surface. Some distortions of conformity, distance, direction, scale, and area always result from this process. Some projections minimize distortions in some of these properties at the expense of maximizing errors in others. Other projection are attempts to only moderately distort all of these properties.

A map projection is conformal if the angles in the original features are preserved in the image on the chart. Over small areas the shapes of objects will be preserved. A line with constant orientation (a rhumb line or loxodrome) will be straight on a conformal projection.
On a conformal projection, the scale is constant in all directions about each point but scale varies from point to point on the map.

For practical nautical navigation only the conformal Mercator projection is of interest. This projection is still used today for nautical charts. The projection was developed by Gerardus Mercator in 1569 as an aid to navigation. It has the special property that loxodromes or rhumb lines are represented as straight lines on the map, which has a fundamental impact upon practical navigation.

The gnomonic projection on the other hand, has the property that great circles appear as straight lines, making it easy to plot the shortest route between two points. Transferred to the Mercator projection, the great circle route will appear as a curved line that can be approximated by straight-line segments. The straight-line segments on the Mercator projection are easier to follow since this only requires following a constant compass direction.


sail022d_A.jpg The "Mercator Projection" is named after the Flemish cartographer Gerard Kremer (1512-1594) , who later adopted the latin name Gerardus Mercator. He published the first maps using his new developed projection in 1569, but only in the beginning of the 17th century, after 1599 when Edward Wright published a detailed explanation of the technique Mercator used, the conformal Mercator Projection became popular among nautical cartographers.

The interesting fact of the Mercator Projection is that the underlying mathematical framework including logarithms (developed by Napier around 1610) calculus (developed by Leibnitz and Newton around 1660) and differential geometry (developed by Gauss around 1800) was not yet available at the time Mercator developed the techniques for his conformal projection.

The conformal Projection

In the 16th century, trading over the new discovered seaways had developed to an important economical factor in western Europe. The compass was the principal navigation instrument and navigation itself was based on dead reckoning. Therefore, it was most convenient to sail tracks of constant heading or sailing along a loxodrome.

With the rapid expansion of sea trading, cartography flourished from the increasingly detailed geographical information. Cartographers not only had to deal with the increasing amount of new data, they also had to think of how to map this data on a chart such that it could be most practically used for sea navigation.

If loxodromes and thus the tracks sailed by the trading ships, would be a straight lines in the nautical chart, this would simplify navigation considerably. At the same time the contours of a coast line on the map must provide a good image of the real coast line. And since bearings are the basis for coastal navigation, nautical maps should also be "angle-true".

This was the problem Mercator was working on during the years around 1560. The requirement that a rhumb line must map to a straight line in the chart, requires that the meridian lines must be mapped to parallel lines on the chart. Together with the "angle-true"- or conformity requirement, the basic grid of a practical nautical chart Mercator was looking for, would consist of an orthogonal grid with meridian lines running parallel bottom-to-top and parallel lines of constant latitude running left-to-right.

If the meridian lines are parallel lines on the chart, but converge to the poles on the globe, the scaling factor between features on the globe and their image on the chart, cannot be the same for all locations on the globe. It was clear to Mercator that to obtain the required characteristics for his practical nautical chart, the scaling factor has to increase with the latitude. In order to obtain conformity, the latitude-depending scaling factor of the chart must be applied in all directions for all points on the chart. So not only the "east-west" directions will be stretched to obtain parallel meridians, also the "north-south" directions must be scaled with the same latitude-depending factor.
In the extreme case, the pole is subject to an infinite degree of distortion since it will be stretched into a line having the same length as the Equator, although it is a point on the globe. Therefore, the image of the poles is infinitely far away from the Equator . As a consequence, the poles will never be visible on a standard Mercator chart. But this has no further practical implications for nautical navigators.

Although the mathematical toolbox to develop an analytical solution to this problem did not yet exist, Mercator managed to develop "scaling tables", with which he was able to construct nautical charts using what we call today a "conformal Mercator projection".

Since there is no geometric projection to produce a Mercator chart, the Mercator Projection is often illustrated as a projection from the center of the earth on a cylinder wrapped around the equator. However, that cylindrical projection is only an approximation of the Mercator projection and it has not the property that Mercator was seeking: the rhumb lines are not straight lines in a cylindrical projection.

The cylindrical projection and the Mercator projection have in common that meridians lines and lines of parallels are straight and intersect orthogonally. Also since the meridian lines don't converge towards the poles (as they do on a globe), the scale of the chart must increase as the latitude increases. The difference to the Mercator projection is that a cylindrical projection has different horizontal and vertical scaling factors and thus it is not conformal.

Nautical charts used today still are Mercator charts. The fact that meridians are parallel makes it easy to measure angles such as headings and bearings from these. But for example in aeronautics, Mercator charts are not used any more because today aeronautic navigation is based on radio navigation. Aircraft pilots use charts which shows great circles as straight lines. Rhumb lines on aeronautical charts are curved lines. Surveyors still use other charts since they are interested in accurate distance and bearings over long distances.

The Mathematical Framework

In mathematics, a projection is a function which takes the coordinates of a point on one surface and transforms them into the coordinates of a point on another surface. More specific in cartography the projection function takes a point from the surface of the globe and transforms it to a point on a plane.

The following variables will be used in the next mathematical equations and functions:

Lat : Latitude in degrees
Lon : Longitude in degrees

X   : east -west  distance on the chart (e.g. in mm)
Y   : north-south distance on the chart (e.g. in mm)
c   : scale of the chart (e.g. in mm/degree)

The conformal Mercator projection

An infinitesimal small square with sides (dLon, dLat) located on the globe at longitude "Lon" and latitude "Lat" has a size on the globe of

dX  = c' * dLon * cos(Lat)
dY  = c' * dLat

with c' the scaling factor between the units of Lon/Lat (°) and the units of X/Y (e.g. km).
However, this transformation maintains conformity only for a small range of Latitude. Indeed, since the spacing of the meridians is not constant but depends on the Latitude, the meridian lines will not run parallel. To obtain parallel meridians, the following scaling must be applied:

c   = c' * cos(Lat)
To obtain conformity, the scaling has to be applied for both directions X and Y:
dX  = c * dLon
dY  = c * dLat / cos(Lat)

The transformation function is obtained by integrating these differential equations. The chart origin (X,Y) = (0,0) is mapped to location (Lon,Lat) = (0°,0°) to resolve the "integration constant":

X   = c * Lon
Y   = c * (ln ( tan ( Lat/2 + 45°))

This is the Mercator conformal projection function, transforming an arbitrary location on the surface of the Earth (Lon,Lat) to a point on a two dimensional chart (X,Y). Longitudes east of the Prime Meridian have positive sign and map to positive X values; longitudes west of the Prime Meridian map to negative values of X. In a similar way, latitudes north of the equator map to positive Y values; latitudes south of the equator have a negative sign and map to negative Y values.

Notice that the above mathematical description is based on a perfect spherical body and does not take into account the ellipticity of the Earth.

The cylindrical projection

The cylindrical projection, which is often used as geometrical illustration of the Mercator projection has the following projection function:

X   = c * Lon
Y   = c * tan(Lat)

This cylindrical projection does not map a loxodrome to a straight line in the chart. In a conformal Mercator projection the scaling factor c is the same for both X and Y dimensions. The cylindrical projection has different scaling factor for X and Y directions and thus is not conformal. This is illustrated in the pictures below for a NW loxodrome:


Loxodrome on a globe

Loxodrome on a chart with a
"Mercator grid"

Loxodrome on a chart with a
"cylinder grid"

The cylindrical projection is nevertheless useful as an approximation of the Mercator projection in a small plotting range (e.g. a couple of degrees of latitude around the location of a vessel).
For small-range plotting areas a chart with an approximate Mercator grid can be constructed as a cylindrical grid in a geometrical way as described in "Worksheets for Lines-of-Position". These plotting sheets are used in the practice of celestial navigation to determine the position from two or more Lines-of-Position.

The gnomonic Projection

The conformal Mercator projection has the great advantage that rhumb lines or loxodromes are mapped to straight lines in the chart. The gnomonic projection produces charts in which great-circle segments are mapped as straight lines. Since a great-circle segment is the shortest track between two points on the globe, gnomonic charts can be advantageously used for nautical navigation to find the shortest route between two ports.
However, the difference in distance between a rhumb line and a great-circle segment connecting the same two points on the globe is only significant for very large distances and higher latitudes (e.g. sailing the Viking's route from Norway to Iceland). Moreover, with simple navigation tools, such as a compass to fix the course, it is rather difficult to sail great-circle segments. So for sailors, gnomonic charts have little practical value.
For aircraft traffic however, the shortest route for long-distance flights is a time and profit issue. And since aircrafts also have sophisticated navigation equipment on board, enabling computer controlled navigation along great-circle segments, gnomonic charts have been very useful for aeronautic navigation.

The gnomonic projection is a geometrical projection obtained as follows: take a point with location (Lon_sp, Lat_sp) in which the plane of projection is tangent with the globe. This point is called the "standard point" or "point of tangency" for the projection.
For any point on the globe with location (Lon,Lat) construct a "projection line" connecting this location with the center of the globe. The location (Lon, Lat) is then mapped to the point where the corresponding "projection line" intersects with the plane of projection.
A great circle is the intersection of a plane through the center of the globe with the surface of the globe. All projection lines of the points of a great circle are contained in the plane which defines the corresponding great circle. The intersection of this plane with the plane of projection is a straight line. Thus great circles map as straight lines for this projection.
Since the gnomonic projection maps antipodal points on the globe to the same point on the chart, it can only be used to map one hemisphere at a time. Only around the standard point, all major characteristics (angles, area, distances, ...) are retained with distortions becoming radially pronounced with increasing distance from the standard point.

The projection function to obtain a gnomonic chart is:
X   = c * ( sin(Lon - Lon_sp) * cos(Lat) ) / cos(a)
Y   = c * ( cos(Lat_sp) * sin(Lat) - sin(Lat_sp) * cos(Lat) * cos(Lon - Lon_sp) ) / cos(a)

with "a" the angular distance of the point (Lat,Lon) to the center of the projection (Lon_sp, Lat_sp):

cos(a) = sin(Lat_sp) * sin(Lat) + cos(Lat_sp) * cos(Lat) * cos(Lon - Lon_sp)

If the location (Lon_sp,Lat_sp) = (0°,0°) is chosen as point of tangency (standard point) the projection function simplifies to:

X   = c * tan(Lon)
Y   = c * tan(Lat) / cos(Lon)

The principle of this gnomonic projection and the resulting chart grid of meridian lines and parallels of constant latitude is illustrated in the pictures below:

The example on the right shows the principle of a gnomonic projection. The projection lines radiate from the center of the globe to the plane of projection, which lies in front of the globe. The point of tangency is located at Lon = 0° and Lat = 0°.

The yellow projection lines are connected to points located on the same parallel (N 30°). The green projection lines are related to some points located on the same meridian (E 30°).
The blue points are located on the surface of the globe and the red points are the corresponding intersection points with the plane of projection on which the gnomonic image of the globe is mapped.

sail022d_F.gif sail022d_G.gif

Loxodrome on a chart with a
"gnomonic grid"

The example of the loxodrome in the picture on the right also illustrates that - even on this scale - there is no real significant difference between a loxodrome and a great-circle segment (the loxodrome is almost a straight line).

Loxodrome distances and courses may also be calulated mathematicaly. This may be usefull e.g. for determining the distance of long tracks covering more than one chart. Also for determining the loxodrome course of transatlantic journeys this mathematical approach may be usefull. Please go to the "notes on loxodrome calculations" to read more on this topic.

To examin the difference between loxodrome tracks and the corresponding great-circle tracks the following interactive pages may be used:

As an example the great-circle distance across the Atlantic Ocean between Cape Town (South Africa; 36°S 17°E) and Montevideo (Uruguay; 37°S 56°W) is 3428 nautical miles. The loxodrome distance is 3521 nautical miles. So the great-circle track is about 100 nautical miles (less than 3%) shorter than the loxodrome track. However, for ocean sailing, taking advantage of favourable winds and currents is usualy of much more importance than sailing the shortest track.

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