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The Mercator Projection

Well, this projection

is one you have doubtless seen many times. I even remember a small atlas, with a red plaid cover, that my mother recieved as a premium with a brand of flour, almost all of the maps of countries and continents in which were in this projection. (If I remember correctly, Canada got a simple conic projection: and, ironically, the world map was in a slightly modified form of the Mercator, where the extreme northern part of the map was divided into round caps.)

If one were to present it in a more balanced form, so that the Equator would be in the middle, it might look like this:

and now one can see the Antarctic coastline as well as the northern tip of Greenland.

This projection was so ubiquitous that it even gave rise to concerns that people were deriving their ideas of what the world was like from it, instead of from the globe.

What made it so popular, and so seductive?

As you can see, as with any cylindrical projection in the normal case, the meridians and parallels are all straight lines. That made it easy to draw. But because of the way the parallels are spaced, every part of the map looks natural. Nothing seems to be streched or twisted, the way it sometimes is on other popular projections.

That is because the Mercator projection is conformal. Why is Greenland so gigantic on the map? Because it has been stretched in the vertical direction exactly enough to compensate for the stretching it must recieve in the horizontal direction on any cylindrical projection.

On the globe,

as this diagram illustrates, the radius of every parallel of latitude is proportional to the cosine of the latitude. This means that the circumference of those parallels follows the same law, and thus the length of a degree of longitude along any parallel of latitude is proportional to the cosine of the latitude as well.

On a cylindrical map, however, the length of a degree of longitude remains the same from the top to the bottom of the map.

Thus, since at a given latitude, horizontal stretching by a factor of one over the cosine of the latitude takes place, vertical stretching of a factor of one over the cosine of the latitude must also be performed.

Thus, the formula for the positions of the parallels in a Mercator projection can be looked up in a table of integrals:

   |     1                  1                                 n         -1
   |  --------, dx = ln( -------- + tan(n) ) = ln( tan( 45° + - ) ) = gd  (n)
   |   cos(x)             cos(n)                              2

And this formula even works when n is negative, representing southern latitudes.

Note the last way of expressing this formula: the height of the parallel for a given latitude, when that latitude is expressed in radians, equals the inverse Gudermannian (also known as the Lambertian) of that latitude.

As the Gudermannian is defined as:

gd(x) = sin  (tanh(x))

the Lambertian, or inverse Gudermannian is also:

  -1          -1
gd  (x) = tanh  (sin(x))

Note that raising the sine function to the -1 power is a notation for the inverse function, or the arc sine, whereas squaring the sine squares the value rather than composing the sine function with itself, here, following conventional mathematical notation, despite the potential for confusion.

Given that:

   2       2
sin x + cos x = 1


    2        2
sinh x - cosh x = -1

or, perhaps, more relevantly,

    2        2
tanh x + sech x = 1

it is possible to establish that the Gudermannian connects the hyperbolic functions to the trigonometric functions in other ways as well. Thus, the Gudermannian can be expressed in all of these ways:

          -1            -1            -1
gd x = sin  tanh x = cos  sech x = tan  sinh x

as well, as, trivially, because the cotangent is the reciprocal of the tangent, and the secant is the reciprocal of the cosine and so on,

          -1            -1            -1
gd x = csc  coth x = sec  cosh x = cot  csch x

which, among other things, allowed the G sub theta scale, in conjunction with trig scales (and another scale for converting from radians to degrees), on the Hemmi 253 slide rule to be used to determine hyperbolic sines, cosines, and tangents.

Because angles are correct at every point on a conformal map, all the compass directions from any point make an equally-spaced circle at each point. On the Mercator, which is a map on a cylindrical projection, in the normal case north is always up. This means that, in the normal case, the Mercator has another important property: compass directions are correct everywhere on it.

This property has made it extremely useful for navigational charts used by ships. A line of constant compass heading is termed a loxodrome, and on the Mercator projection, all loxodromes are straight lines throughout their length.

However, it is a great circle that is the equivalent to a straight line on the globe, the shortest distance between two points. Going north or south, or east or west along the equator, the two are the same. Going east or west anywhere else, parallels of latitude are the loxodromes, and they are small circles, not great ones. Loxodromes in other directions are a type of spiral on the surface of the sphere, somewhat analogous to a logarithmic spiral.

But that doesn't mean the Mercator projection is not useful in other cases.

In the oblique case, for example, a Mercator projection like this:

can serve as the source for an attractive map of North and South America, since, as can be seen from the map above, those continents are both crossing the map's equator, the line of minimum distortion, which is shown as a line in red on the map.

On the right, we see the portion of that map containing the Americas, rotated 60 degrees clockwise, The Oxford atlas contained a conformal map of the Americas on a similar projection.

A map projection which provided a further reduction in error for the Americas was based on the Lambert Conformal Conic, using two conic projections which were then stiched together by means of a small section which departed from strict conformality. This projection, the Bipolar Oblique Conic Conformal, has been deservedly acclaimed, but because of the complexity of dealing with that small area where its two segments are joined, I have not tried to describe it here.

The Transverse Mercator, also called the Gauss Conformal, is useful for star maps of the kind that show how the sky appears at a particular time on a particular day of the year, because the resulting map can be equally useful for people who live at any latitude.

It is fitting that Karl Friedrich Gauss concerned himself with a projection useful for astronomy, as he is also the designer of the Orthoscopic eyepiece, a design still considered ideal for planetary astronomy, due to its extremely low distortion.

The Transverse Mercator is also currently popular for very large-scale maps. Why is this, when it would seem that the simplest possible projections would suffice, such as the simple conic, the Mercator in normal case, or even the Azimuthal Equidistant?

The reason for the popularity of the Transverse Mercator, and for the previous popularity, as an approximation, of the Polyconic and related projections is because even on a large-scale map, there are slight errors because the map is flat and the Earth is round.

If the Azimuthal Equidistant were used for each map, at least each map would be on the same projection as each other map, and so one set of tables for the corrections needed could be used.

On the other hand, it is also desirable to be able, as far as possible, to allow adjacent maps to be joined together. This can be done in one direction. The simple conic will allow this horizontally, but it is a different projection for each standard parallel.

The Transverse Mercator combines both advantages: maps made in that projection can join up vertically, and any part of the Earth can be made to pass through its standard meridian, so every place can be mapped using the same projection. In practice, the actual printed maps used are segments of a larger map, sharing a common standard meridian. This increases error, but the correction tables needed still only need to cover a very narrow horizontal strip, and doing so allows a grid coordinate system to be used over a larger area.

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