Benjamin Høyer: Cartography

Closing Comments

In addition to the perspective projection types, there are any number of purely conventional projection types. Conventional here does not mean that they are normal or common-sense. It means they follow mathematically defined rules which relate the coordinates of a point on the sphere to one on the plane.

Moving away from projections that can be described entirely with geometry gives freedom to preserve any quantity, and control the severity of distortions in the other quantities. However, almost invariably, these distortions will vary according to wheren the map you are. For example, in Bonne's projection, shape and area distortion are far worse near the left and right edges of the map.

Interrupted Projections

To counter this problem the projection are often broken into parts, generating an interrupted projection. Conveniently, the Earth's continents are spread in such a way that the interruptions can be positioned to split oceans, with minimal interference to land. A common three-way split is shown in Figure 31, where North and South America, Europe and Africa, and Asia and Australia are shown together.

Figure 31
http://members.shaw.ca/quadibloc/maps/mps0401.htm

Some interesting projections have been produced using azimuthal projections six times to produce a cubic map of the world. Figure 32 shows a development of thr gnomonic projection onto a cube, with the central face centred on 0°N 0°E. Figure 33 describes the difficulties of finding the great circle route (straight line in the gnomonic projection) across the interruptions on the cubic gnomonic projection in Figure 32.

Figure 32
The Study of Map Projections, J A Steers (University of London Press, 1956) p48

Figure 33
The Study of Map Projections, J A Steers (University of London Press, 1956) p49

Haversine Formula

Before the age of computers, the haversine was considered one of the most important trigonometric functions. In fact, the earliest surviving trigonometric table from the fifth century consisted only of the sine and versed sine, with angles ranging from 0° to 90° in 3.75° increments.

The name comes from half of the versine, which in turn comes from the Latin, sinus versus, meaning flipped sine. It is defined as in Figure 44 or by the formula

haversin(φ) = ½ versin(φ) =sin² (½ φ)       (6.1)

The haversine formula was an equation important to sailing since it gave the great-circle distance between any two points. It also gave the bearing from one to the other. Since the bearing is not constant along a great circle, the formula would be applied multiply to find a number of waypoints, giving a route which could be easily followed by a compass, like in Figure 22.

The formula is derived from the spherical law of cosines and is written in terms of the latitude of the two points and their difference in latitude and longitude, thus

h = haversin(d/R) = haversin(Δφ) + cos(φ₁) cos(φ₂) haversine(Δλ)     (6.2)

Then

d = haversin^-1(h) = 2R arcsin(√h).

Errors are introduced when the two points are approximately on the opposite side of the sphere. In this case, care must be taken that h does not exceed 1 since the arcsin in the expression for d gives complex results when h > 1. Also, errors are introduced in the assumption that the Earth is a sphere rather than an ellipsoid. However, as mentioned before, these errors are small and typically on the order of 0.1%. An equivalent set of equations does exist that takes the planet s ellipticity into account, called Vincenty's formulae.

Craig Retroazimuthal or Mecca Projection

A common use of the haversine formula today could potentially be for Muslims to find the direction of their qibla. However, many maps and tools exist for them to find correct direction for any given location without having to resort to a calculator. One such tool is the Craig Retroazimuthal
Projection.

The word retroazimuthal refers to its property that the bearing to the centre from any point is correct, which is precisely the concern of Muslims looking to pray in the direction of Mecca. It was developed by James Ireland Craig in 1909 while he was working as a cartographer in Egypt. It manages to keep the directional properties of the Hammer retroazimuthal projection, but without the extreme shape distortion associated with that projection (compare Figures 35 and the central part of Figure 36). The projection is defined by the mapping equations

x = λ
y = λ /sin λ (sin φ cos λ - tan φ₀ cos φ).     (6.3)

Note: for locations on the meridian λ = 0, λ/sin λ = 1 should be taken, the continuous limit.

Figure 35
http://www.mapthematics.com/Projections.html

Figure 36
http://www.mapthematics.com/Projections.html

Nowadays many tools exist online, often based on Google Maps, which give an interactive solution to the qibla problem. For example, applying one tool to St Andrews yields picture in Figure 37.

Figure 37
http://www.qiblalocator.com/