How maps are made, and the first conformal map

It is impossible to map a sphere onto a plane faithfully. A *map*, in the mathematical sense, is a one-to-one correspondence of points on two different surfaces. A *faithful* map preserves shapes and distances. To preserve local shapes, it is sufficient for the angle between any two lines in one surface to be mapped into the same angle in the other. A map satisfying this requirement is called *conformal*. Maps may differ in *scale*, but the scale of a faithful map must be uniform. Small areas on the earth may be approximately faithfully represented by plane maps, but for larger areas this is impossible, and for the whole earth it is a difficult problem even for approximate maps. A globe, however, is a faithful map of the earth's surface if it is carefully made. The law governing the transformation of the coordinates of a point on one surface to its coordinates in its map is called the *map projection*.

A close approximation to a faithful map is a conformal map with a scale that varies regularly from place to place. In fact, such maps are generally simply called *conformal* maps. Another kind of map, frequently found in atlases, drops the requirement of conformality for other criteria, such as equality of areas. All such maps distort shapes, and so are useless for any quantitative purpose. The length of a line drawn on them has no special significance, since the scale must depend on direction in a non-conformal map. Distances and directions scaled off such maps are approximate at best, and there is no way to make them more exact. A typical atlas (Hammond Odyssey Atlas of the World) says optimistically that the non-conformal Robinson projection it uses shows "the whole earth with relatively true shapes and reasonably equal areas." A glance at the map shows that continental shapes are greatly distorted. Alaska is noticeably sheared, and trans-Pacific relations are obscured. Such maps are good only for looking and decoration, not for any serious purpose. How bad they are can be discovered by comparing them with a globe.

I wanted the distance from Mt. McKinley in Alaska, longitude 151° W, latitude 63° N, to Seattle, longitude 122.33° W, latitude 47.61° N. By trigonometry, the great circle distance is 2459 km. Measurement on a 30-cm globe gave 2486 km, and on a Lambert map of North America, 2485 km. The Robinson projection gave 3907 km, an error of 60%. Moreover, the great-circle route on the globe and Lambert map hugged the British Columbia coast, while on the Robinson projection the straight line veered far out into the Gulf of Alaska (and was not a great-circle route at all). This is striking evidence of the inferiority of non-conformal projections.

The *scale* of a map is the ratio betweeen the distance on the map and the actual distance represented, although sometimes quoted as the reciprocal and called the Representative Fraction, RF. 1:48 000 or 1/48 000 means that one millimetre on the map represents 48 metres on the ground. A series of Army maps at 1:250000, about 4 inches to the mile, covers the United States. A *small scale* map, such as 1:1 000 000, represents an area as seen from a great distance, while a *large scale* map, such as 1:24 000, represents an area seen from closer up. The scale of a map usually varies over the area of the map, but on a good conformal map it does not vary much from the quoted value. It is easy to find the scale of a globe by measuring its diameter.

The location of any point on the globe can be specified by its geographical coordinates, the latitude and longitude. The equator is a great circle halfway between the poles, and meridians are great circles running from pole to pole. Great circles are straight lines on the sphere. Longitude is measured east and west from the Greenwich meridian as an angular distance along the equator where the meridian intersects it, from 0° to ±180°. Latitude is measured from the equator northward and southward from 0° to ±90°. A line of constant longitude is called a *meridian*, and a line of constant latitude a *parallel*. Meridians are straight lines (great circles), but parallels are circles of a radius that decreases towards the poles. The equator is the only parallel that is a great circle. The earth is slightly oblate, but this complication will be ignored here. It can be taken care of in the mathematics if great accuracy is required. The equatorial radius of the earth is 6 378 140 metres, the polar radius 6 356 775 metres, and the flattening 1/297.257. If you could see the whole earth, it would appear perfectly spherical.

Claudius Ptolemy, of the 2nd century, collected latitudes and longitudes of many locations, and plotted them as rectangular coordinates. Latitudes could be determined with some accuracy, but longitudes were always uncertain until time could be measured accurately, which was relatively recently, as late as the 18th century. Ptolemy's map, with later additions and extensions, was used for over a thousand years, and even guided Columbus to America. The problem of creating maps that could be used for navigation became critical in the 16th century with the voyages of discovery, since Ptolemy's map was not well-adapted to compass navigation and did not include enough of the earth.

Gerhardus Mercator (1512-1594), whose Flemish name was Gerhard Krämer, solved the problem in 1568 with his map of the world on a new principle. The meridians and parallels on his map are straight lines crossing at right angles, as on Ptolemy's map, but the parallels are more widely spaced at higher latitudes north and south, and the polar regions are not represented at all. Mercator reasoned that since the meridians approached one another as one approached the poles, while the distance between them was represented by a constant distance on the map, the perpendicular distance should increase proportionally, to make the scales equal in all directions. We recognize this as the requirement for a conformal map. Mercator had no calculus, as we have, to work out the projection exactly, but he could measure the distances between meridians on his globe and draw his map accordingly, using what we would now call numerical integration. I have only deduced this from the evidence available to me, and do not know at first-hand what Mercator's reasoning was.

The principle of the Mercator map is illustrated at the left. Two areas between two meridians are shown, one at the equator and the other at a high latitude. The equatorial area is chosen to be square, with sides represented by unity. Each side corresponds to the same small increment of longitude or latitude, say 1°. The area at high latitude is narrower for the same 1° increment; let this distance be x. It is just as high, however, for the same increment of latitude, so it is not a square, but a rectangle. On the map at the right, two meridians are drawn, any convenient distance apart, to represent a difference of 1° of longitude. On the earth, this distance is about 69 miles or 111 km. The square on the sphere maps into the square between the meridians. In the map, the meridians do not converge, so when the rectangle is brought over from the sphere, it is just as high, but its width is only x < 1. To make it fit between the meridians, it must be enlarged by a factor 1/x as shown. The rectangle has the same shape, but it is larger. If this is done repeatedly, beginning at the equator, the coordinate net for the Mercator map is obtained. The construction is approximate, but if the angle increment is chosen small enough, the result will be satisfactory. If this explanation is understood, it will be easy to follow the calculus later.

The Mercator map is a conformal map with the scale decreasing toward the poles. Shapes are accurately rendered (unless they are so large that the scale varies considerably across them). A straight line drawn on the map crosses any meridian at the same angle, which is its (true) *compass bearing*. That is, if you travel always keeping the compass needle fixed, you will travel along a straight line on a Mercator map, called a *loxodrome* or a *rhumb line*. The compass, of course, points to magnetic north, not geographical north, and must be corrected for this *declination*. Alternatively, if you draw an arbitrary course on the map, it can be approximated by a series of straight lines, each with a fixed bearing. Such a map is of great use to navigators. Mercator's map made the magnetic compass a practical tool for accurate navigation, instead of just a rough and uncertain guide. The compass is really of little use without accurate maps. The scale can be determined accurately for any point on a Mercator map, so that distances can be scaled off precisely. The central meridian of the map can be taken as any meridian. Indeed, the map can be cut up along meridians and reassembled with any meridian at the center.

The Mercator is, then, a *working* map, not just a decoration. It can be based on any great circle on the sphere, not just the equator, and is then called a *transverse Mercator projection*. The area for some distance to each side of the central great circle is accurately and conformally represented, though meridians are not straight lines. Such a projection is used for U. S. state surveys, and for military maps. It shares the stage with the Lambert Conformal map, another excellent map projection, which we may discuss elsewhere. The Mercator map was the first map that could represent most of the whole world at one time, sacrificing only the polar regions and uniformity of scale. From it, many schoolchildren formed the impression that Canada was much larger than the United States, when in fact they are about equal in area.

The word "projection" comes from a graphic way to illustrate the relations between points on the globe and corresponding points on a map. These methods do indeed give maps, but not conformal ones except in a special case, and are not the basis for practical map projections. However, the projections may help to illustrate the subject. It should be clearly understood, however, that maps are not usually made in this way. Suppose we wrap a plane around a sphere so that it makes a cylinder. After points have been projected on the cylinder, it can be unwrapped to make a flat map, which is what we desire. The Mercator projection is similar to this, so it is called a *cylindrical* projection. There are also *conic* and *plane* projections.

In the diagram at the right, the circle is the meridian containing point P, with latitude φ. The line at the right represents a generator of the cylinder surrounding the sphere. For mathematical purposes, we assume that the radius of the sphere, to which all distances are proportional, is unity. A line from the center O through P intersects the cylinder in point A. A line perpendicular to the axis N-S through P intersects the cylinder in point C. Each of these projections produces a map of the sphere, but neither map is conformal. Some point B at a distance z from the equatorial plane is the projection that will make a conformal map. We must find z as a function of φ, which will be the mathematical representation of the Mercator projection.

The radius QP determines the distance on the sphere for a small change in longitude Δθ. This short distance is actually along a parallel of latitude, but if short enough cannot be distinguished from a segment of a great circle--that is, a straight line. This distance is cos φ dθ, while the meridional distance is dφ. The scale of the map (map/actual) vertically is dz/dφ, while the horizontal scale is dθ/cos φ dθ. If these scales are equal, then dz = dφ/cos φ. This is a differential equation for z(φ) that can be solved by simple integration, which is carried out at the left. The final form is customary, containing only one trigonometric function to look up, and which can be easily evaluated by logarithms. This is not so important these days, but it is still a convenient expression.

The scale of the map is given by 1/cos φ, as we found above. The area scale is the square of this. For example, the average latitude of Alaska is about 65°, so the map scale is 2.366, relative to the scale at the equator. Areas are multiplied by 5.599 times. The average latitude of the United States is about 40°, so the map scale is 1.305 the equatorial scale, and areas are multiplied by 1.7. Therefore, Alaska looks about 1/3 greater in area than it should relative to the coterminous United States on a Mercator map. Conversely, equatorial areas seem smaller than they should.

For the central projection, that gave point A above, z = tan &phi, and for the lateral projection, z = sin φ for point C. Since tan φ > ln tan(φ/2 + π/4) > sin φ, the Mercator projection is a mean between them. The geometric projection that does give a conformal map is shown at the right. This *stereographic projection* can be used for maps of the polar regions. The plane of projection is shown as the equatorial plane, but any parallel plane will work as well. Mathematically, the projection is r = cos φ / (1 + sin φ) = tan (π/4 - φ/2). The meridians are lines radiating from a point representing the pole, and parallels of latitude are circles, which is quite correct. The projection can be proved to be conformal by finding the scales in perpendicular directions at any point. Stereographic projection is extensively used in crystallography.

In addition to projections onto cylinders and planes, which have suggested the Mercator and sterographic projections, the sphere can also be projected on a cone, which can be flattened out to a plane. Again, a geometric central or parallel projection does not give a conformal map, but a conformal map can be defined mathematically in which the scale is the same on any two parallels of latitude. This Lambert Conformal Projection is excellent for zonal regions of the globe, and can be extended indefinitely far east and west. It deserves a page to itself, so is only mentioned here for comparison with the cylindrical and planar projections. It is treated in its own page.

The familiar Mercator map is based on the great circle of the equator. Any great circle can be taken as the basis, in particular a meridian. When this is done, the projection is called the Transverse Mercator. The coordinate transformation from latitude and longitude with respect to the equator to latitude and longituded with respect to the chosen meridian (they all are the same) is shown at the right. Solution of the shaded right-angled spherical triangle gives the desired relations, which are shown on the diagram. The Mercator projection is then carried out on the new coordinates λ' and φ' in the usual way.

A Transverse Mercator coordinate net showing meridians and parallels appears at the left. The orthogonal intersections show that it is conformal. Neither meridians nor parallels are straight lines, or any simple curve. At the center of the map, meridians and parallels are almost a rectangular net. The map is used only in this area, centered on a certain longitude and latitude. The basic map has unity scale on the standard meridian, but the scale can be changed slightly to make it smaller than unity on the meridian, and unity a certain distance east and west of the meridian, so that the scale is closer to unity over a wider band of the map.

The Transverse Mercator projection is used as a basis for the Universal Transverse Mercator (UTM) grid system for military maps. It is easy to cover any relatively small area anywhere on the globe with this system, though not maps showing a large area, when the great changes in scale would be objectionable. A grid system overlays a rectangular grid on the map, to which points are referred instead of using longitude and latitude. The earth between 80°S and 80°N is divided into quadrilateral zones 8° N-S and 6° E-W, numbered 1-60 eastward beginning at 180° and C-X (I and O omitted) south to north. These are divided into 100 000-m squares designated by two letters. The principle of stating a UTM grid reference is shown at the right. The zone designation, 12S, is added if references cover a wide area. This reference locates point P, the village of Red Rock on the New Mexico-Arizona border in the Navajo Reservation, to within 1000 metres. More precise grid coordinates, XR735526, locates Red Rock to 100 metres. A special L-shaped ruler facilitates accurate reading of the grid coordinates. One used by the Army had scales of 1:25000 and 1:50000, with metres on one side and yards on the other. In order to make a grid reference, a map is required, of course, such as the one in the References. I have not investigated in detail how these grids and squares are made to fit together, and what approximations are involved. An arbitrary square grid could easily be superimposed on any map, but making it correspond to distance with any accuracy is a more difficult question.

The earth is actually a spheroid, as discussed in more detail in Lambert's Map, and this must be taken into account in accurate mapping. A reference spheroid is specified by its equatorial diameter a, and its polar flattening f. A meridional cross-section of the earth is an ellipse of eccentricity e given by e^{2} = 2/f - 1/f^{2}. The IAU spheroid has a = 6378140 m and 1/f = 298.257, which gives e = 0.0818. The latitude is the angle between vertical at any point and the equatorial plane.

It is not very difficult to work out the Mercator projection for the spheroid by integrating the equation that results when the meridional scale is set equal to the longitudinal scale at any point. I quote only the result, which is y = -a log[|tan(ψ/2)|{(1 + e cos(ψ))/(1 - e cos(ψ))}^{e/2}]. The scale factor is SF = sin(ψ)/[1 - e^{2}cos(ψ)]^{1/2}. In these equations, ψ is the colatitude, 90° - φ. It is very easy for a computer program to evaluate these formulas.

As a test, we can see how well these formulas work on the earth. The distance between latitudes 40° and 41° is 111 042.39 m (about 69 miles), from geodetic tables. The formulas give y = 4 838 450.27 and 4 984 280.98, respectively, for a difference of 145 830.71. The scale at 41° is 0.76710697, and at 40°, 0.75580016, so the average scale is 0.76145356. Multiplying the scale by the northing difference, we get 111 043.31 m, differing only 0.92 m from the tabular value. This is pretty good agreement (1 part in 111,000) for such a simple calculation.

As we said above, a straight line on a Mercator map is called a *loxodrome*, Greek for "slanting course." It is a course of constant compass heading, very convenient for navigation. It represents the shortest distance only when approximately north-south or east-west, since the scale of a Mercator map is not constant, but changes with latitude. Any long course is divided into a series of loxodromes. A Mercator map is not good for comparing regions at different distances from the equator, but always gives an accurate appreciation of the shape of limited areas. It is not applicable to polar regions. No good way to represent the whole globe on a plane map has yet been found. Look at the various attempts displayed in atlases; the best tear the surface of the globe into disconnected regions.

My 1960 Hammond world atlas includes a very nice separate Mercator projection of the world. It reaches to about ±83 ° latitude. The central meridian is at longitude 82° W, which puts the Americas in the centre but splits Asia at eastern China. There is a scale diagram showing the scale in statute miles at latitudes from 0° to 60°. By considering the longitude scale, we find that the parameter a is 181 mm. Now we can calculate the position on the map of any point if we know its latitude and longitude. There is a table of geographic coordinates of U. S. cities in the annual almanac currently issued by Time (with Information Please).

Let's consider a voyage from San Francisco (37° 47' N, 122° 26' W) to Anchorage (61° 13' N, 149° 54' W). From the projection formula, and using a = 181 mm, we find that San Francisco should be at z = 0.7132a or 129 mm from the equator, and Anchorage at z = 1.3602a or 246 mm. Measurement on the map verifies these figures. The loxodrome is the hypotenuse of a triangle with legs 246 - 129 = 117 mm and 87 mm. The length of this course is 146 mm, and its bearing is N 36.6° W. These results can be compared with those found by drawing a line on the map, and the agreement is close. The average latitude of the course is 51°, for which the scale diagram gives 85mm = 1200 miles. This makes the approximate length of the course 2118 miles, or 3388 km.

It is easy to find the scale as a function of latitude by differentiating the projection equation. We find dz = adφ/cos φ. This is the scale along the meridian, but since the mapping is conformal, it is the scale in any direction. At 51°, dz = 1.589adφ. Since a represents the radius of the earth, the scale at the equator is 181 mm = 6378 km, or 35.2 km/mm. At 51°, the scale is then 35.2/1.589 = 22.2 km/mm. The length of the course is then 22.2 x 146 = 3241 km. This isn't far from the very approximate result we read from the scale diagram.

With a piece of string and a globe, we can estimate the great circle course from San Francisco to Anchorage. It goes a little closer to the continent, but is really not too far from the loxodrome. The great circle distance from Anchorage to San Francisco, assuming the earth to be a sphere of radius 6378 km, is 3228 km by calculation. This is only 13 km shorter than the loxodrome.

J. N. Wilford, *The Mapmakers* (New York: Vintage Books, 1981) is a historical survey of cartography from early times to the present.

A. Robinson, R. Sale and J. Morrison, *Elements of Cartography*, 4th ed. (New York: John Wiley and Sons, 1978). A good undergraduate text for geography students, but thorougly non-mathematical. Concentrates on the physical map itself as a graphic aid.

U.S.G.S. (Army Map Service) 1:250000 series, Shiprock Quadrangle, sheet NJ12-12, 1969. Any map of this series will do.

FM21-25, *Elementary Map and Aerial Photograph Reading* (War Department, 15 August 1944). An excellent little manual, written for the World War II soldier, and showing practices of that period, including the use of a pocket compass. Geographical coordinates and map projections are not included, of course, but a lot of practical information on using grids and thrust lines is.

My HP-48 program, which I call AD, to calculate great circle distance (as an angle) is {ROT - COS ROT ROT COS LASTARG SIN ROT COS LASTARG SIN ROT * ROT ROT * ROT * + ACOS}. It expects the stack to be set up as LAT1 LONG1 LAT2 LONG2 (top to bottom). Convert angles to decimal with ←TIME function HMS→, and the final decimal angle to radians with MTH REAL NXT NXT D→R. Finally, multiply by the radius of the earth.

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Composed by J. B. Calvert

Created 25 August 2002

Last revised 30 April 2009