## Shape of the Earth

The earth is approximately a sphere, but with a varied and irregular surface. One of the basic aims of the science of geodesy is to determine the position of points on (as well as above or below) the earth’s surface, in a way which is unique and unambiguous. Although no simple mathematical model exists to cope with the variations in the earth’s true shape, historical attempts to define its size and shape discovered that it approximates to an oblate spheroid (that is a sphere slightly squashed at the poles), or an ellipse rotated about the semi-minor axis which is aligned to the axis of rotation of the earth (an ellipsoid). In fact, points on the earth’s surface deviate by up to 9 km from the best fitting global ellipsoid.

## Geoid Model

So what is the true shape of the earth? The geoid is defined as a surface on which the earth’s attractive (i.e. gravitational) forces are everywhere equal, i.e. a gravimetric equipotential surface. This may be visualised conceptually as a surface which coincides with mean sea level (imagining there was no land), where the effects of non-gravitational forces, such as tides, currents and metreological effects, are removed. However, there are local gravitational anomalies to this simplistic concept, due to land mass, noticed particularly in mountainous areas, and these can distort the shape of the geoid locally.

The geoid is of fundamental importance in determining positions on the earth’s surface as most measurements are made with reference to this surface. For instance, heights are referred to mean sea level (which is effectively the geoid), and many measurement devices, such as theodolites, use gravity to determine directions. Furthermore, satellite systems operate within an environment directly influenced by gravity. The geoid is not a simple mathematical surface (although it can be modelled), but deviates by up to 100m from an ellipsoid, largely due to variations in gravity around the globe.

## Reference Ellipsoid

Because the ellipsoid is a good approximation to the shape of the geoid, and it is simple to define mathematically, it has been used in classical geodesy for over 200 years to provide a figure of the earth on which positions may be given in terms of latitude, longitude and height above the ellipsoidal surface. The ellipsoid thus used is termed a reference ellipsoid. As stated before, the shape of the geoid varies around the globe, therefore different sized ellipsoids have been used for different regions. Each is chosen to fit the geoid as closely as measurement technologies and computational abilities allowed at the time they were established. For example, an ellipsoid which provides a good fit of the geoid over the whole globe is not necessarily the most suitable for North America, and neither would be the most appropriate for Ireland (see the diagram below for an exaggerated depiction).

Exaggerated diagram of regional ellipsoids

This shows an imaginary section through the earth, with the regional ellipsoids positioned to closely fit the geoids of the country concerned. In a similar way, the best fitting global ellipsoid would not necessarily be the most appropriate for either region.

## Geodetic Datum

There are many different ellipsoids on which positions may be expressed. The size, shape and positioning of the ellipsoidal reference system with respect to the area of interest is largely arbitrary, and determined in different ways around the globe. The defining parameters of such a reference system are known as the geodetic datum. The geodetic datum may be defined by the following constants:

• the size and shape of the ellipsoid, usually expressed as the semi-major axis (a) and the flattening (f) or eccentricity squared (e2). There are a number of techniques used to determine the best fit ellipsoid for an area. Historically, triangulation was used in Britain and Ireland;
• the direction of the minor axis of the ellipsoid. This is classically defined as being parallel to the mean spin axis of the earth, and achieved by comparing the observed astronomic bearing of a line (say in a triangulation) with its calculated ellipsoidal bearing, satisfying the Laplace condition, and adjusting the triangulation network as appropriate;
• the position of its centre, either implied by adopting a geodetic latitude and longitude (F, ?) and geoid / ellipsoid separation (N) at one, or more points (datum stations), or in absolute terms with reference to the centre of mass of the earth;
• the zero of longitude (conventionally the Greenwich Meridian).

The manner in which the Geodetic Datum is defined varies from country to country (or region to region), usually through survey observations, adoption of international standards, or acceptance of some form of historical convention.

## Height Datum

Because the geoid is an irregular shape, its surface is not generally parallel to the ellipsoidal surface. Therefore it is usual to fix the geoid at one location, usually some reference mark at which height above mean sea level has been determined, and refer heights to this point for practical purposes – this is known as the height or vertical datum.

Although the relationship between the geoid and ellipsoid is known at this point, and may be known at certain other points, the separation is not constant and furthermore can vary considerably, depending upon the nature of the geoid in the area of interest. Therefore some model of the variation may be required in order to determine the separation elsewhere. By choosing the best fitting ellipsoid this separation can, in certain circumstances, be ignored. However, with global ellipsoids, or in areas of significant terrain variation, the separation and variation can be significant, particularly when transforming positions between reference systems. In these circumstances a geoid model is important.

## Cartesian Co-ordinates

Positions may be given in absolute terms, relative to the earth’s centre of mass, or an assumed centre (as implied by a geodetic datum). With Cartesian co-ordinates a position is defined in 3 dimensional space by an X, Y, Z co-ordinate triplet. The Z axis passes through the centre of the earth (or reference ellipsoid) and the poles, the X axis through the centre and the Greenwich meridian, and the Y axis at right angles to these. Other parameters may define this system, but are not directly relevant here.

It is important to realise that the centre of a Cartesian reference system for an adopted ellipsoid may not be the ‘true’ or adopted centre of mass of the earth – the latter is often used for global reference systems (as used by the Global Positioning System, GPS). Furthermore, the direction of the Z axis may differ. This has led to one of the major requirements of geodesy and surveying today, which is how to relate global to local referencing systems so that positions in one may be expressed in terms of the other (and vice-versa). This issue is not dealt with here, except to state that it is necessary to move from the ellipsoidal system to the local cartesian reference system before applying some translations in X, Y and Z, and possibly scale change and rotations around these axes as well, to move from the local system to global.

The cartesian reference system is particularly useful as calculations are simpler to perform (no knowledge of spherical geometry being required). However, the relationship between positions on the earth’s surface are difficult to visualise in this system, and this is particularly important in navigation (for instance); the concept of height is also unclear.

## Geodetic (Ellipsoidal) Co-ordinates

With Geodetic (Ellipsoidal) co-ordinates the position of a given point (P) may be measured on the earth’s surface and given a latitude (F) and a longitude (?) in terms of the reference ellipsoid (see diagram 1 below) by projecting it onto the ellipsoid’s surface along a line in the direction which is perpendicular to the ellipsoid surface (the normal) – see diagram 2 below).

Diagram 1: Ellipsoidal Reference System

Diagram 2: The relationship between the ellipsoid, the geoid, land surface, normal and true vertical

It should be noted that the direction of the true vertical (direction of gravity) may be slightly different from the normal, and this difference in direction is termed the deviation of the vertical, which for most practical purposes may be ignored. The distance along the normal to the ellipsoid is termed the ellipsoidal height (usually designated by the letter H), and fixes the point in 3 dimensional space. As noted before, the height above sea level (in geodesy termed the orthometric height, and designated by h) is the most useful height for practical purposes, and is usually measured by spirit levelling. The separation between the ellipsoid and geoid along the normal is generally known as the geoid-ellipsoid separation, (designated conventionally N), and is an important element in the computational process. Thus, positions are given in terms of latitude, longitude and ellipsoidal heights.

## Plane (Grid) Co-ordinates

Having established a 3-Dimensional co-ordinate reference system, it remains to depict the position of points of interest onto a flat surface, e.g. a piece of paper. In general, it is usual to determine the position of some reference points in terms of the ellipsoidal reference system by measurement (such as triangulation), before projecting these onto a plane co-ordinate system and carrying out the survey of topographical detail in this simpler system. Projecting co-ordinates from a curved surface (latitude, F and longitude, ?) on to a plane surface will cause some distortion, but this is minimised by choosing the most appropriate projection for the area concerned, and the features to be depicted.
It is usual to depict features from the earth’s surface on paper, or some other two dimensional medium. This is achieved by mathematically projecting geographical co-ordinates onto a plane. Positions can then be expressed in terms of eastings and northings.

• Eastings are the distance, in metres, in an easterly direction from some origin.
• Northings are the distance, in metres, in a northerly direction from some origin.

Diagram 1: Eastings and Northings

There are many projections which can be used, and these may be seen in any world atlas. However, because the areas to be projected in a world atlas are large the distortions are unavoidable and clearly evident. Engineering and cadastral maps are generally of smaller areas depicted at scales larger than 1:50 000, and as these forms of map are often the basis for further measurement, it is necessary to keep distortions to a minimum. The property of retaining shape and scale is known as orthomorphism, or conformality.

Transverse Mercator Projection

One projection with the properties of orthomorphism is known as the Transverse Mercator, or Gauss Conformal projection. Although conceptually simple, the mathematics involved are less so (in order to ensure orthomorphism).
A cylinder of a specified radius is wrapped around the reference ellipsoid so that its circumference touches the ellipsoid along a chosen meridian (line of longitude); (see diagram 2 below). The scale of the projected area is therefore correct along the chosen meridian. The radius of the cylinder will match the radius of the reference ellipsoid at a specified point. This provides the Origin of the projection (i.e. the line of longitude at which the cylinder touches, and the latitude of the reference ellipsoid where the radii match). The cylinder is then ‘unwrapped’, providing the flat surface of the map. The origin is generally chosen to be central to the area of interest, so that the distortions away from the origin are minimised.

Diagram 2: Concept of Transverse Mercator Projection

Scale increases away from the origin in an uniform way, so that scale at any point on the grid away from the central meridian (where it is uniform) may be corrected by applying a scale factor to determine true scale. In this way, for short lines, true ground distances may be obtained by measurement off the map.