## Benjamin Høyer: Cartography

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## Mathematical foreword

Assumptions, simplifications and conventions

Before we can begin to define, describe and analyse projections in common use today, we need to take a look process of considering the real Earth as a mathematical object.

The Earth as a Sphere

The actual shape of the Earth is highly irregular. In fact, defining what is taken to be the surface is a tricky topic in itself. For example, the surface of constant gravitation is known as the geoid. Smoothing and simplifying this surface, the Earth can be described as pear-shaped, an ovoid. This has been shown theoretically and measured experimentally. Most commonly, however, the Earth described as ellipsoidal, the minor axis running from pole to pole. This simplification introduces an error of only about 115km on the Earth's circumference, amounting to 0.1% error.
(Equatorial radius 7927 miles, polar radius 7920 miles. Hence, 7927/7920 = 1.00088)
Many other factors in mapmaking will likely introduce greater errors, and given that most cartographic methods aim to create maps of, say, one metre across, this error translates to just 0.1mm in print, this is an acceptable simplification.

Calculating on a Sphere

The sensible choice of coordinate system for cartography is a modified spherical coordinate system. Since the Earth is reduced to a perfect sphere (i.e. every point is a constant R away from the centre), each point can be located using only two angles, reducing the dimension of the problem by one.

Since a sphere itself has no natural start or end from which to measure, reference lines must be defined by convention. Starting from the axis of rotation, poles can be defined and the locus of points equidistant from both poles describe the Equator. This line allows us to describe a point's extent in the north/south direction. To define the east/west extent, the 0° is set by convention as going (approximately) though Greenwich: the Prime Meridian.

From these lines, angles can be taken and a point on the surface can be identified uniquely.

Figure 1
Showing longitude λ and latitude φ measurements.

Naming Conventions

Spherical polar coordinates differ in definition in cartographic mathematics in comparison to standard mathematics. The difference is simply a change in origin and range for each of the angles, as defined in Figure 1. In mathematics, angles are measured in radians, while geography tends to use degrees.

In geography, the quantity φ is called latitude and λ is called longitude. Each of these has associated with it parallels where their values are constant. Lines of constant φ (parallels of latitude) define concentric circles around the Earth, with the north (φ>0) or south (φ<0) pole as the centre. Lines of constant λ (longitude) are called Meridians and are all the same length: half the circumference of the Earth.

Special reference points on Earth and their names:

 φ = 0 Equator φ = ½π = 90° North Pole φ = -½π = -90° South Pole λ = 0 Prime Meridian (Greenwich) λ = ±π = ±180° International Date Line

Great and Small Circles

The equator and all meridians are great circles. That is to say,

• they cut the Earth into two equal halves;
• they are the largest possible circles that can be drawn;
• the share their centre with the Earth sphere.
A great circle can be drawn between any two points on Earth and this path will represent the shortest path along the surface, analogous to a straight line in the plane (the line of minimum curvature). Parallels of latitude, other than the equator and poles, are not great circles: they are small circles. A small circle's diameter will be smaller than the of its parent sphere, but the shortest route between two points on the circle will always pass through its interior, along a great circle.

The Problem of Map Projection

In his 1828, Carl Friedrich Gauss produced a proof called Theorema Egregium. This proof states that any two surfaces with the same Gaussian curvatures are mathematically equivalent. While this has some extremely helpful consequences (studies on complicated shapes can be carried out on simpler shapes of the same curvature), it does mean that no map projection can ever be accurate: the Earth's curvature, R2, is not equals to plane's, 0. Thus, the object of map projection is only to convey as much accurate information of the original globe on a flat piece of paper. There is any number of possible separate ways to do this, but they can be categorised according to

• whether they are true perspective projections or not;
• what surface they project onto.
A projection is true perspective if it can be projected in the literal sense. This is best visualised with a light bulb. Can the projection be carried out by shining light "through  the surface of the Earth, drawing around the shadow on the projection surface?

Figure 2

The categories for projection surfaces are as follows.

Azimuthal Projections
This type is the most simple and intuitive: the globe is projected onto a flat surface (plane). It immediately becomes clear that little more than one hemisphere can be mapped using a true perspective azimuthal projection. The projection surface is positioned so as to be tangential to the Earth at the projection's centre. The direction taken from a point to the centre of the projection will show the true bearing, or azimuth: hence its name.

Cylindrical Projections
For this type, the Earth is place in the centre of a cylinder. The projection is onto the wall of the cylinder, which is then cut along a line and rolled out flat. The line cut is arbitrary but is often chosen to avoid cutting landmasses (often the anti- meridian is used, keeping the prime meridian in the centre of the projected map).

Conic Projections
Simple conic projections align the vertex of the cone with the axis of rotation and project onto its surface from the centre of the Earth. The cone can be set to touch the Earth along a single small circle, or to intersect the Earth twice.

Each projection surface (plane, cylinder and cone) can easily be transformed into a flat plane, the cone and cylinder being split down the side and unrolled into a plane.

Preserved Quantities

True perspective projections can easily be adjusted and improved to minimise distortion -- or eradicate it entirely -- for certain measurements. Three separate qualities which can be maintained to an extent, and at most one can be maintained in any one projection.

Orthomorphism -- Shape
Shape is often a concern in general maps. We want to get an idea for the shape of countries and continents, and the oceans surrounding them. What we are really saying here, is that the relation between points should be preserved as much as
possible -- the relative distance between them and the angle between them. Shape conservation becomes a question of preserving angles locally. The scale in each direction must be the same at every point, although not necessarily constant at all points, as defined in (2.1) and Figure 3.

sr = sθ for all r, θ or equivalently sx = sy for all x, y       (2.1)

Figure 3

Conformal -- Equal-Area

Having an idea for the shape of a landmass give of some information, but we know very little if we don't know its size in relation to other landmasses. An equal-area map will preserve differences in area between countries and continents. Equal- area maps are of great political importance: they show countries in their true relative sizes, avoiding misleading compressing of equatorial countries. They are also used in, for example, showing relative vegetation coverage. For a map to be conformal, the product of the scales must be unity at all points.

sr × sθ= 1 for all r, θ or equivalently sx × sy = 1 for all x, y       (2.2)

Figure 4

Bearing

For air and sea navigation, having a map which shows the correct bearing between points is clearly helpful. While this will not generally give the great circle (shortest) route, it is the easiest route to follow by compass, as the bearing is constant. Generally maps do not give the correct bearing between points. In some cases, they do give the great circle route, which can then be traced out on a bearing- preserving map for navigation purposes.

Figure 5
Maps, Their Uses and Construction, G J Morrison (Edward Stanford, 1902) p26

Another important feature of a map is its scale. Part of the reason to use a map is convenience of size -- the Earth must be reduced to something more manageable. The scale on the map cannot be constant, which is what gives rise to the distortions listed above: it is the relationship between the longitude and latitude scales which affect the accuracy.

Mathematically, projections are defined entirely by how the parallels of longitude and latitude are translated to the page, whether true perspective or otherwise. This mesh is known as the graticule. Knowing the location of borders and other points, a picture of the Earth can then be recreated according to their longitude/latitude positions.

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B Høyer June 2009