Meridian

1The impossible problem

A globe represents the Earth without distortion, but a globe cannot be carried in a pocket, printed in a book, or displayed on a screen. For those purposes the spherical surface must be rendered on a flat one, and the rendering is never faithful. The reason is not a want of ingenuity. It is a theorem.

A sphere and a plane differ in a property that mathematicians call intrinsic curvature, and the difference cannot be removed by any bending that does not also stretch. An orange peel pressed flat must tear or crease; it cannot be made to lie flat while keeping every distance upon it unchanged. Carl Friedrich Gauss proved the general statement in 1827. His result, which he named the Theorema Egregium, the remarkable theorem, established that curvature is intrinsic to a surface and is preserved under any bending that does not stretch. Because the sphere and the plane have different curvature, no stretch-free correspondence between them exists, and every flat map is therefore a stretched representation of the Earth.

A map projection is a precise statement of how the stretching is to be done: a rule assigning to each latitude and longitude on the globe a point on the sheet. Since distortion cannot be eliminated, the rule can only distribute it. A projection may keep angles faithful, so that small shapes are correct, or keep areas faithful, so that no region is inflated relative to another, or keep certain distances faithful. It cannot keep all of them. The design of a projection is the deliberate choice of which fidelity to keep and which to surrender, and this document is an account of those choices.

2A history of map projections: a literature review

The study of projections passed through three phases: a long period of projections devised individually for particular needs, a shorter period in which their design was placed on a systematic mathematical footing, and a modern period of cataloguing and of projections designed by explicit compromise. This section traces that course, with the usual emphasis on the limitation each advance was meant to remove.

Projection is an ancient practice; Ptolemy described conic constructions in his Geography in the second century. For more than a millennium afterward, projections were devised one at a time, each for an immediate purpose. The most consequential of them was published by Gerardus Mercator in 1569 [1], and its purpose was navigation. Mercator's projection has the property that a course of constant compass bearing appears as a straight line, so that a navigator could rule a line between two ports and read off a single heading to sail. The projection answered that need so completely that its severe distortion of area, which inflates the high latitudes without bound, was accepted for centuries as the price.

The individual projections accumulated without a unifying theory until Johann Heinrich Lambert supplied one in 1772 [2]. Lambert's contribution was method. Rather than devise a projection and then examine its properties, he began from a property to be preserved, conformality or equal area, and derived the projection that preserved it. From this systematic approach came a whole set of projections still in constant use, among them the Lambert conformal conic and the Lambert azimuthal equal-area. The design of projections became a branch of mathematics rather than a craft.

A deeper question remained unanswered even so: was the distortion that every projection exhibited a removable defect or a necessary one? Gauss settled it in 1827 with the theorem of the previous section [3]. The Theorema Egregium proved the distortion necessary, and in doing so it changed the object of the field. The goal could no longer be a faithful map, because none exists. It had to be the understanding and the deliberate control of an unavoidable distortion. What that goal required was a way to measure distortion precisely, and the measure was provided by Nicolas Auguste Tissot, whose construction, developed from the 1850s and given its full treatment in 1881 [4], is the indicatrix described in section three. Tissot's indicatrix made the comparison of projections a quantitative matter rather than a question of judgement.

The modern period is one of cataloguing and of explicit compromise. John Snyder's Map Projections: A Working Manual of 1987 [5] collected the formulae, properties and appropriate uses of the projections in practical use into a single reference, and his later history, Flattening the Earth [6], recorded how each had come about. The period also produced projections designed to preserve no property exactly. Arthur Robinson's projection of 1963 [7] was constructed by adjusting a table of values until the whole-world map simply looked right, trading every mathematical fidelity for an overall balance of appearance. The history thus closes where it began, with projections devised for a purpose, but the purpose is now stated openly and the distortion is understood, measured and chosen rather than merely suffered.

3Tissot's indicatrix

To compare projections one must measure distortion, and to measure it one needs an instrument that works at a single point, because distortion varies across a map. Tissot's indicatrix is that instrument. Consider a very small circle drawn on the surface of the globe. On the globe every such circle is identical. Run the circle through a projection and it is in general no longer a circle but an ellipse, and the shape, size and orientation of that ellipse record exactly what the projection has done to the neighbourhood of that point.

The figure shows the simplest projection, the equirectangular, on which longitude becomes horizontal distance and latitude becomes vertical distance in direct proportion. Its indicatrices tell a plain story. Near the equator they are nearly circular and nearly true to size. Toward the poles they stretch wider and wider, because every meridian has been pried apart to the full width of the equator while on the globe the meridians converge to a point. Two quantities summarise each ellipse. Its area, compared with the area of the undistorted circle, gives the projection's area distortion at that point. Its departure from a circular shape gives the angular distortion. The probe reports both. With this instrument the four families of the following sections can be compared on a common and exact footing.

4Cylindrical projections

A cylindrical projection may be pictured as the globe wrapped in a cylinder, its features cast outward onto the cylinder, and the cylinder then unrolled. Meridians become equally spaced vertical lines and parallels become horizontal lines. The one freedom left to the designer is the spacing of the parallels, the vertical distance between the horizontal lines, and that single choice fixes the projection's character.

Mercator made a definite demand of that choice: the map was to be conformal, preserving angles everywhere, so that any small shape appeared true. Conformality requires that the vertical stretching at each latitude match exactly the horizontal stretching the cylinder already imposes. Near the poles the horizontal stretching grows without bound, so the parallels must be spaced ever further apart to keep pace.

The indicatrices state the bargain exactly. On the Mercator projection every indicatrix is a circle, which is conformality made visible: a circle has no long axis, so no direction is favoured and angles are preserved. The probe confirms that the angular distortion is zero everywhere. The circles' size, however, grows enormously toward the poles. Shape has been kept by surrendering area without limit, and this is the origin of the projection's most criticised feature, the appearance of Greenland as the rival of Africa though Africa is the larger by a factor of fourteen. Greenland is not drawn too large by error; it sits where the indicatrix has swollen. The projection nonetheless endured for four centuries, because for the navigator's single purpose, a straight line of constant bearing, nothing surpassed it.

5Conic projections

A cylinder touches the globe along a single line, the equator, and so a cylindrical projection serves the tropics well and the rest of the world poorly. A cone can be set on the globe so that it touches, or cuts through, at a latitude of the designer's choosing, which moves the region of low distortion to wherever it is wanted.

A conic projection casts the globe onto such a cone and unrolls the cone into a fan. Meridians become straight lines radiating from a point and parallels become concentric arcs. The cone is usually chosen to cut the globe along two standard parallels, two latitudes at which the map touches the sphere and the scale is exactly true. Between the standard parallels the map is slightly compressed; outside them it is slightly stretched; and across the whole band they straddle the distortion stays small.

The figure's slider moves the standard parallel, and the probe confirms the effect: along the standard parallels the indicatrices are nearly true circles of nearly unit area, and the distortion grows only as the probe moves well above or below them. The practical lesson is to aim the projection, choosing standard parallels that bracket the region of interest. This is why conic projections are the basis of the official maps of mid-latitude countries. The national surveys of the United States, of much of Europe, and of the broad east-west expanses of Russia and Canada rest on conic projections, because a cone can be tuned to follow one wide band of latitude closely.

6Azimuthal projections

Collapsing the cone until it touches the globe at only a single point gives a plane resting on the globe, and a projection onto that plane is called azimuthal. Parallels become concentric circles about the point of contact and meridians become straight spokes radiating from it. The figure uses the North Pole as the point of contact.

The defining property is named in the word azimuth, which means a compass direction. On an azimuthal projection the direction from the point of contact to any other place is correct: a straight line drawn on the map from the centre points along the true bearing to its destination. The particular azimuthal projection shown, the azimuthal equidistant, adds a second fidelity. Distances measured from the centre are also true to scale, so a ruler laid from the centre reads genuine ground distance. The whole Earth fits within the figure, the opposite pole spread around the outer rim.

The probe shows what is kept and what is spent. Near the centre the indicatrix is a near-perfect circle of near-unit area, so close to the point of contact little is distorted. Moving outward, the ellipses stretch along the parallels, sideways to the radial spokes, while remaining true along the spokes themselves. Direction and radial distance are honoured; shape across the radial direction is given up, increasingly so with distance from the centre. The projection is the natural choice wherever the controlling question is direction and distance from one place, as on the range rings of a control tower's display or the antenna chart of a radio operator, and the polar azimuthal equidistant map is the projection on the flag of the United Nations.

7Equal-area projections

Mercator preserved angles and allowed area to grow without limit. The opposite choice is to preserve area exactly and to allow shape to bend. For a map of the whole world this is frequently the sounder choice, and a projection that makes it is called equal-area.

An equal-area projection guarantees that any region occupies a portion of the map proportional to its true area on the globe. No country can be inflated and none can be diminished. The figure shows the Mollweide projection, published by Karl Mollweide in 1805, an ellipse twice as wide as it is tall, with meridians curving as smooth arcs to the poles.

The indicatrices report the consequence. Their area, wherever the probe is placed, stays close to that of the undistorted circle, which is the guarantee being kept. Their shape, however, changes constantly: round near the centre of the map, increasingly leaning and flattened toward the corners. An equal-area projection spends angular fidelity to buy honest area, the exact reverse of Mercator's bargain. This is why equal-area projections are the standard for any map whose purpose is to show how a quantity is distributed, population or rainfall or forest cover, since a distortion of area would be read by the eye as a feature of the data itself.

8Conclusion

The study of map projections begins from an impossibility. Gauss proved that a sphere cannot be flattened without distortion, so the question is never how to avoid distortion but how to distribute it. Tissot's indicatrix made the distribution measurable, turning each projection into a definite pattern of area distortion and angular distortion across the map. Against that common measure the four families take their places. Cylindrical projections, and the conformal Mercator, keep angles for the navigator. Conic projections hold a chosen band of mid-latitudes closely for the national survey. Azimuthal projections keep direction and distance from a single point. Equal-area projections keep size for the honest comparison of distributions.

The literature review placed these families in their history, from the purpose-built projections of Ptolemy and Mercator, through Lambert's systematic theory and Gauss's proof that the distortion is necessary, to Tissot's measure and the modern catalogue. A reader who has followed the argument here holds the conclusion the whole field rests on. There is no correct map of the world, only a map correct for a question, and the mapmaker's skill lies not in escaping distortion but in choosing it.