Benjamin Høyer: Cartography

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Cylindrical Projections

In a cylindrical projection, the projection surface is wrapped around the sphere forming a an open- ended cylinder. The surface features on the sphere are then projected, perspective or otherwise, onto the inside of the cylinder. Finally the cylinder is split along a seam and rolled out flat. When unrolled, the cylinder will have a width of 2πR, the circumference of the sphere.

Unless mathematical adjustment is carried out afterwards, we can immediately see that there will be extreme scale enlargements at high latitudes either side of the equator (in the normal or equatorial case), since all parallels are represented by lines of length 2πR. While this cannot be reduced, it can readily be compensated for, since the natural way to present the mapping equations is in cartesian coordinates, and they can easily be stretched independently on one another, unlike the polar mappings for the azimuthal projections.

True cylindrical projections have these three points in common (in their equatorial cases),

1. meridians project to straight lines,
2. parallels project to straight lines,
3. meridians and parallels cross at right angles.
Equirectangular Projection or Plate Carrée

This projection is the cylindrical analogue of the azimuthal equidistant projection mentioned previously. It can be thought of as a rod through the centre of the sphere, emitting light perpendicular to its axis, but distances along meridians are preserved, as in Figure 17: OP=OP'.

Figure 17

Given that the parallels are projected adjusted to preserve length along meridians, the meridian scale is 1. The parallel scale however is not -- each parallel, save the equator, is elongated onto the wall of the cylinder. This immediately tells us the projection can neither be area nor shape preserving.

The parallel scale is found by comparing the true length of parallels to the projected length. Each parallel is projected to a constant 2πR, while each has a true length variable on its latitude. The result is clearly an elongation.

Parallel scale factor = DP' / DP = 2πR / 2πR cos φ = sec φ      (4.1)

The result of this projection is a rectangular graticule, twice as wide as it is tall. This owes the fact that the height represents half the circle -½π < φ < ½π and the width represents the full revolution, -π < λ < π. If drawn using equal angular divisions, the parallels will form squares, hence the name Plate Carrée. It is worth remembering that the lines on the graticule represent separations of equal angles, not distance: a parallel angular distance of 15° at 60°N will have arc length half of that at the equator. Despite the right-angled intersections of the parallels, the projection is neither area, shape nor straight-line preserving. In Figure 18, θ =α8 δ.

Figure 18
Latitude and longitude in (square) 30° sections.

Since neither of the three main quantities are preserved on this map it seems of little use. It could perfectly well be used so long at the latitudinal extent is sensibly restricted. For example, within ±30° of the equator, areas and parallel lengths have an error of about 15%. However, other projections exist preserving shape and area better, without much additional work. The next two maps each correct either the area distortion or the shape distortion inherent to the plate carrée. Since the parallel scale is quite simple, it is not difficult to find a way to stretch the meridian scale so that either

1. the parallel distortion is equalled or,
2. the parallel distortion is cancelled out by inversion.
Lambert's or Cylindrical Equal-Area Projection

This projection is able to show the Earth in its entirety, preserving correct area. With area preservation, the total area of the projected surface must equal that of the projection surface. Hence the surface area of a sphere must be represented precisely on a rectangle (which is then rolled up into a cylinder).

R2 = h × (2πR) with h = 2R = d       (4.2)

Thus, the height of the final map is equal to the height, or diameter, of the sphere. This implies that Lambert's projection is the cylindrical analogue to the orthographic projection mentioned earlier. The map can be produced by perspective, the light rays radiating perpendicular to the central axis of the sphere.

Figure 19

Equation (4.2) shows the projection overall, on average, preserves area. To be equal area, all parts on the globe must be equal in area to their projected counterparts. This can be confirmed by investigating "belts  of surface between two latitudes.

Figure 20

Their area may be calculated to be

A = R2 (sin φ2 - sin φ1)(λ2 - λ1).      (4.3)

This formula states the area between two lines of constant latitude is simply the circumferential length of the band, multiplied by the perpendicular distance between them. Setting the longitudinal width to a full revolution (π to -π), the formula simplifies to

A = 2πR(sin φ2 - sin φ1)      (4.4)

Rearranging, we get

A = 2πR × R (sin φ2 - sin φ1)    h = R (sin φ2 - sin φ1)  vertical weight of belt    A = 2πRh    (4.5)

where h is the vertical height of the belt. Clearly, this is also the calculation for the belt's projection on the cylinder. Hence the projection preserves areas across variations in latitude and, by circular symmetry, all areas. Its conformal property can be confirmed by scale analysis.

Parallel scale: DP' / DP = 2πR /2π cosφ = 1 / cos φ    (4.6)

The meridian scale is found by analysis identical to the orthographic meridian scale -- see (3.28) to (3.35) -- but the initial length used is related to sine rather than cosine, i.e. (4.7), since the projection is orthogonal.

P'Q' = R sin φ2 - R sin φ1     (4.7)

Analogously to (3.35), this yields the result

P'Q' / PQ = (-2 cos φ) × (-½) = cos φ      (4.8)

Checking area preservation using the formula given in (2.2) we confirm

sφ × sλ = 1 / cos φ × cos φ = 1      (4.9)

Clearly the orthomorphic condition (2.1) of scale parity is not satisfied.

Mercator's or Cylindrical Orthomorphic Projection

This is undoubtedly the most common of all global map projections in use today. In almost all cases, people's image of the globe and the relation between landmasses on it comes from the Mercator projection, and so too do their misconceptions. While it does have some very convenient features, its use as a general wall map is misleading. It is notoriously bad at preserving area and in recent years moves have been made towards projections which address this. The Mercator's projection works in a similar way to Lambert's, in that it compensates for the exaggerations in the parallel scale by adjusting the meridian scale. While Lambert's projection compresses the meridians to preserve area, Mercator's stretches them to preserve shape. In other words, parallel and meridian scales are equal satisfying (2.1). Since the parallel scale is the same for all true cylindrical scales, we can immediately assert

Parallel scale factor  sφ = sλ meridian scale factor   sφ = 1 / cos φ = sλ    (4.10)

Parallel stretching is accompanied by an equal meridian stretching and so the area stretching can be expressed simply as 1 / cos2φ. This extreme area exaggeration is what makes the map so unsuitable for comparing landmasses. The table below gives an idea for the errors involved. Above latitudes of about 30° the errors begin to become significant.

 Latitude Length increase Area increase 0° 0% 0% 15° 4% 7% 30° 15% 33% 45° 41% 100% 60° 100% 300% 75° 286% 1393% 80° 476% 3216% 85° 1047% 13065%
Table 1

A commonly cited example of the area exaggeration is the comparison of Greenland and South America. Since South America ranges from about 10°N to 50°S and Greenland ranges from about 60°, their relative areas are not represented well at all. In fact, they appear about the same size, while in fact Greenland is about ten times smaller than South America. Since the shape and direction are true for every point, straight lines represent lines of constant bearing on the real sphere. These straight lines are called loxodromes. This property makes it invaluable for navigation, both on sea and in the air. Over short distances and small seas, the Mercator projection's straight lines are reasonably close to the great circle routes6. Over larger routes, however, the line of constant bearing is far from the shortest route.

Figure 21
Maps, Their Uses and Construction, G J Morrison (Edward Stanford, 1902) p29

When used for long journeys, for example transatlantic flights, the great circle route is obviously desirable. However, it is difficult to follow a great circle course, since the bearing is not constant. Historically the method to get around was to first map the great circle course on a great-circle preserving map, like the gnomonic, and then transfer to the Mercator using waypoints. This breaks the great circle route down into series of constant bearing sections.

Figure 22
Map Projections, H S Hoblin (Edward Arnold, 1969) p23

It is also possible to avoid having to use two maps, one to determine the great circle route and one to determine the loxodrome parts. Common in the days before the electronic computer was able to perform complication calculations quickly, sailors would determine the direction to set off in for the great circle route to another port using the haversine formula. More information will be given in the final chapter.

As in the case of azimuthal projections, any number of variations are possible if we depart from perspective projections. Common modifications are to lessen the extent of parallel stretching. If, for example, the parallels have their length preserved, we arrive at the Sanson-Flamsteed or Sinusoidal Projection.

Sanson-Flamsteed or Sinusoidal Projection

Figure 23
http://www.progonos.com/furuti/MapProj/Dither/ProjTbl/Img/tn-Sinusoidal.png

The projection is easily constructed and is often used for maps of Africa and South America. However, due to its awkward shape distortion at the edges, it is not much use for the whole globe.

The length of the central meridian, i.e. the height of the map, is true as is the length of the equator. The parallels also preserve their length, but the other meridians are all elongated. Thus, the half- projection (e.g. eastern hemisphere) can be enclosed in a rectangle of equal sides, Rπ.

Figure 24

It is known at the sinusoidal map because the curved meridians either side of the meridians are sine curves of various amplitude. This can be seen from observing that the perpendicular distance from the equator at any point is given by

CD = r = R cos φ      (4.11)

Galls' Projection

This projection introduces another possible adjustment in cylindrical projections. The cylinder this time has radius half that of the sphere, cutting the sphere along two small circles, 45°N and S. The projection then occurs stereographically, i.e. from the equatorial antipode for each point. The angle at the projection source (denoted δ previously) ranges from -45° to +45°, as in the diagram below.

Figure 25

Since this projection touches its projection plane along two small circles, there are two lines of true representation, ±45°. This projection is commonly used for whole-world projections, for example in Philip's Modern World Atlas. It makes a good job of minimising exaggerations polewards, but since no property is preserved, it is not usual for anything other than getting a general idea of the Earth. The radius of the cylinder can be adjusted, changing the latitudes at which the map is accurate. Note that between the two incident latitudes, areas are compressed, whereas polewards they are enlarged.

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B Høyer June 2009