# Introduction

I love maps. But how do they work? This blog post examines some of the physics behind maps and how they work.

# Representing the Earth on a Piece of Paper

If we assume the Earth is a sphere for now. Its not really, its quite a complex shape but for now we will keep things simple. More on its non-spherical nature later.

So a globe can be made to be quite a good approximation of the Earth. However, its not possible to represent the 3-D Earth on a 2-D piece of paper without distortion. You can see how this is the case by taking something spherical like an orange or an old football and trying to flatten out the surface layer (the peel in the case of the orange, or the football once deflated). The only way to flatten the surface layer is to tear it and / or stretch or deform it.

So, to get a representation of the globe on a sheet of paper a projection is required that will cause some sort of distortion. So compromises need to be made. The distorted properties are:

- Distances
- Directions
- Shapes
- Areas

There are many ways that a map projection can be done. These can be visualised as wrapping the Earth with a piece of paper in some way to mark all the detail on the piece of paper and then flattening out the piece of paper.

## Mercator Projection

Gerardus Mercator, 1512 – 1594, was a Dutch Cartographer who came up with a world map in 1569.

The Mercator projection is a conical projection. Conical projections preserve angles (this is called conformality) but distort other properties like areas. The cone touches the Earth, for example, at the Equator. Nearer the point where the cone touches the earth, the distortion is smallest; further away it gets progressively bigger. So, typically on World maps the distortion gets bigger nearer the poles.

Here is a typical Mercator projection map of the world. The classic distortion is the size of Greenland being much bigger than Australia where in reality, Australia is more than 3.5 times the size of Greenland!

This graphic flips between the Mercator projection and actual areas of countries.

The reason Mercator would have chosen this projection is because of navigation. With a conformal (equal angles) map, if you traverse a straight line from point A to point B, you will traverse a course of equal angle to the meridians on the Earth. A line of equal angles is called a Rhumb Line or Loxodrome. So, to sail from one side of an ocean to another, draw a line on the map and measure the angle, then sail that constant angle and you will reach your desired destination. This is obviously easy to do from a navigation perspective.

With a projection that is not conformal, Rhumb lines are not straight lines. Or, put another way, if you follow a straight line on a non-conformal map, you will need to constantly change your bearing as you get nearer to your destination. Possible, but far harder from a navigation perspective.

So for a standard Mercator projection centred on the Equator, the map is very useful around the equator but less so nearer the poles. Lambert realised that if you rotated the piece of paper relative so the Earth, so it touched a meridian of longitude then you could produce a map accurate around that meridian. This is the Transverse Mercator Projection and is useful for mapping areas with a north south extend, like for instance South America.

## Gnomic Projection

Marine maps are often Mercator projections. But to travel large distances it is usually better to take the shortest route. The shortest route between two points is a straight line, right? Well, not if you are travelling on the Earth (unless you plan to tunnel through the Earth to your destination). To travel from A to B on the Earth’s surface, you will travel a curved line along the Earth’s surface. The shortest curved surface route would be the Great Circle going through the start and end points. A great circle is any circle on the surface of the Earth centred at the centre of the Earth (assuming the Earth to be a sphere for a moment).

Rhumb lines on a Mercator projection map are not Great Circles so are not the shortest way to travel from any given start point to an end point. A “planar” projection map called a Gnomic projection is defined to represent all Great Circles as straight lines and so drawing a straight line from start point to end point will result in following a Great Circle on the Earth’s surface which is the shortest distance between the start and end points (on the Earth’s surface). Note, to steer this course will require constant bearing adjustments.

## Lambert Conformal Conic Projection

The Lambert Conformal Conic projection is similar to the Mercator projection but is a conic, rather than cylindrical, projection. The imaginary cone touches the Earth twice. Being conformal, angles are preserved so Rhumb lines are straight lines on this projection (like Mercator).

However, by touching the cone on the two extremities of the area to be mapped, distortion within the mapped area is less than with a Mercator projection, and the Rhumb lines are nearer Great Circles than on a Mercator projection.

The Lambert Conformal Conic projection is often used for Aeronautical charts (as above).

## More Projections

Many projections have been produced for specific purposes. A large list is here.

## Globe

A globe, of course, suffers none of the distortions of a 2-dimensional map, but is obviously more difficult to carry about and impractical at a more detailed level.

# The Shape of the Earth

The Earth is obviously a complex shape with mountains and valleys, land and sea. It is best approximated by an oblate ellipsoid, that is a sphere that has been squashed slightly at the poles, so it is wider at the equator.

So a point on the equator is approximately 21km (13 miles) further from the Earth’s centre than the North or South Pole.

By convention, mountains are measured as height above sea level (even if they are nowhere near the sea). An alternative would be to measure their distance from the centre of the Earth. Everest is 28º North, so clearly at a disadvantage compared to mountains near the equator. In fact, Chimborazo, a peak near the equator in the Andes turns out to be the furthest point from the centre of the Earth (even though it is not the highest peak in the Andes!). Everest does not make the top 20 mountains using this method of height calculation.

## Ellipsoids

So the best simple shape to use to model the Earth is an ellipsoid but it is only an approximation. Depending on the intended use it may make sense to use one ellipsoid over another. For example, the best fit global ellipsoid may be a relatively good fit in some areas but relatively poor in others so typically country or regional maps will use an ellipsoid that makes sense for that country or region.

Globally, the most common ellipsoid in use is the World Geodetic System 1984, WGS-84. GPS uses this. In the UK, it is common to use the Airy ellipsoid (calculated by George Airy in 1830), for example, this is what Ordnance Survey uses.

Here are some common ellipsoids:

## Geoid

By convention, “height” is measured with respect to sea level. But what sea level? Tides rise and fall, the moon influences the height of the high tides at different times of the year, etc. So some conventions are required?

Typically, an average sea level is chosen as height = zero. The shape of the Geoid is defined as a surface perpendicular to the Earth’s field of gravity, having a constant gravitational potential. Gravity generally acts towards the centre of the Earth but due to differences in the composition of the Earth there are variations on both small and large scales meaning that gravity does not always point at the centre of the Earth. The shape of the Geoid is therefore as complex as that of the Earth’s surface, but usually approximated by an ellipsoid.

### Global Geoid

The Global Geoid is usually defined with height relative to the average sea level across all the world’s oceans. This leads to an ellipsoid accurate to about 200m across the whole world, quite good but not that accurate.

### Local Geoid

In the UK, the Local Geoid height is measured relative to a tide-gauge at Newlyn in Cornwall. The seas around the UK are below the global average and the Newlyn gauge measures about 80cm below the Global Geoid! The seas are not all at the same level, due to temperature variation, tides, winds and water impurities.

This also means that measuring sea level at Newlyn from year to year will give different results (and that is before global warming comes into play!). The Ordnance Survey defines Ordnance Datum Newlyn as the mean sea level at Newlyn between 1915 and 1921. It is still used on maps and marked as ODN, and all heights on Ordnance Survey maps are relative to ODN.

## Datum

A datum is a convention used to define a coordinate system and reference points in order to describe a point on the Earth. So a datum would typically reference an ellipsoid, a coordinate system and an origin of that coordinate system. It is a little confusing because sometimes the datum and its ellipsoid have been given the same name, e.g. WGS84.

The datum may refer to a globe, which is unprojected or to a particular map projection that specifies how to translate from the 3D ellipsoid to a 2D map, e.g. a Mercator projection.

Lets look at a couple of examples:

### World Geodetic System 1984, WGS84

GPS uses WGS84 as it is a global system with the satellites orbiting the centre of the Earth on elliptical paths, so basing measurements from the centre of the Earth is quite convenient. This datum is unprojected. WGS84 is defined by the following:

- The WGS84 Cartesian axes and ellipsoid are geocentric; that is, their origin is the centre of mass of the Earth including oceans and atmosphere.
- Their orientation (that is, the directions of the axes, and hence the orientation of the ellipsoid equator and prime meridian of zero longitude) coincided with the equator and prime meridian of the Bureau Internationale de l’Heure at the moment in time midnight on New Year’s Eve 1983.
- Since this time, the orientation of the axes and ellipsoid has changed such that the average motion of the crustal plates relative to the ellipsoid is zero. This ensures that the Z-axis of the WGS84 datum coincides with the International Reference Pole, and that the prime meridian of the ellipsoid (that is, the plane containing the Z and X Cartesian axes) coincides with the International Reference Meridian.
- The shape and size of the WGS84 biaxial ellipsoid is defined by the semi-major axis length a = 6378137.000 metres, and the reciprocal of flattening 1/f = 298.257223563.

### UK Datum OSGB36

The National Grid defines the UK into a series of 100km x 100km squares each given a 2 letter code. The datum specifies a Transverse Mercator Projection. The origin (called True Origin) is defined at 49°N 2°W. To remove the need for negative values west of the origin, a False Origin is defined 400km west and 100km north of the True Origin.

Any point in the UK can be defined to a 1 square metre accuracy as:

- 2 character grid square
- 5 digit “easting”. Number of metres east of the southwest corner of the square
- 5 digit “northing”. Number of metres north of the southwest corner of the square.

# GPS

Following a map on a GPS is relatively straightforward. If you need to take a position reference from a GPS and compare it to, say, a map, then it is important that the GPS is configured in the same way as the map. Typically, this would involve configuring:

- Datum
- Ellipsoid
- Grid System

So, in the UK you would set the OSGB36 datum, Airy ellipsoid and UK National Grid system.