# Chapter 3.  Shape of the Earth and its practical simplification

There are several approaches to define the shape of the Earth. In our study, we need one that is in a form of a function. This function should give just one value to given spheric or ellipsoidal coordinates. This value can be a length of a radius from the center to our point, or an elevation over a specific theoretical surface.

An obvious selection would be the border of the solid Earth and the hydrosphere with the atmosphere. However, this approach immediately raises some problems of definition: should the ’solid’ vegetation be a part of the shape of our planet? How could we handle the buildings or the floating icebergs?

Still, if we could solve these above problems, there still is another theoretical one: this definition does not result an unambiguous function. In case of the caves or the over-bent slopes there are several altitude values connected to a specific horizontal location. The shape of the border of the phases should be somewhat smoothed.

The field of the gravity force offers exactly these kinds of smoothed surfaces. The geoid (’Earth-like’) shape of the Earth can be described by a specific level surface of this force field. There are infinite numbers of level surfaces, so we choose the one that fits the best to the mean sea level. From this setup we obtain the less precise, however very imaginable definition of the geoid: the continuation of the sea level beneath the continents. Let’s see, how this picture was formed in the history and how could we use it in the practical surveying.

## 3.1 Change of the assumed shape of the Earth in the science

The ancient Greeks were aware of the sphere-like shape of our planet. The famous experiment of Erathostenes, when in the exact time of the summer solstice (so, at the same time) the angles of the Sun elevation were measured at different geographical latitudes, to estimate the radius of the Earth, is well known. However, the accuracy of the estimation, concerning the technology of that age, is considerably good.

Although the science of the medieval Europe considered the Greeks as its ancestors, they thought that the Earth is flat. Beliefs, like ’end of the world’, the answer to the question: what location we got if we go a lot to a constant direction at a flat surface, were derived from this.

The results of the 15th and 16th century navigation, especially the circumnavigation of the small fleet of Magellan (1520-21) made this view of the world obsolete. However the Church accepted this only slowly, the idea of the sphere-like Earth was again the governing one.

There were several observations that questioned the real ideal spherical shape. In the 17th century, the accuracy of the time measuring was increased by the pendulum clock. The precisely set pendulum clocks could reproduce the today’s noon from the yesterday’s one with an error of 1-3 seconds. If such a correctly set up clock was transferred to considerably different latitude – e.g. from Paris to the French Guyana – higher errors, sometimes more than a minute long ones, were occurred. This is because the period of the pendulum is controlled by the gravitational acceleration, that is, according to there observations, obviously varies with the latitude. Paris is closer to the mass center of our planet than the French Guyana is, thus the ideal spherical shape of the Earth must be somewhat distorted, the radius is a function of the latitude, and the real shape is like an ellipsoid of revolution.

Distorted but in which direction? Elongated or flattened? The polar or the equatorial radius is longer? Perhaps nowadays it is a bit surprising but this debate lasted several decades, fought by astronomers, geodesists, mathematicians and physicists. Finally, the angular measurements, brokered by the French Academy of Sciences, settled it. In Lapland, at high latitudes, and in Peru, at low altitudes, they measured the distances of meridian lines between points where the culmination height of a star was different by one arc degree. The answer was obvious: the Earth is flattened; the polar radius is shorter than the equatorial one.

The flattened ellipsoid of revolution can be exactly defined by two figures, as it was shown in Chapter 2. Traditionally, one of them is the semi-major axis, the equatorial radius, gives the size of the ellipsoid. The other figure, either the semi-minor axis or the flattening or the eccentricity, gives the shape of the ellipsoid. The time of the invention of this concept, the authors usually gave the inverse flattening. This figure describes the ratio between the semi-major axis and the difference between the semi-major and semi-minor axes.

At the end of the 1700s and in the first half of the 1800s, several ellipsoids were published, as the better and better approximations of the shape of the Earth. These ellipsoids are referred to as the publishing scholar’s name and the year of the publication, e.g. the Zách 1806 ellipsoid means the ellipsoid size-shape pair described be the Hungarian astronomer-geodesist Ferenc Zách in 1806.

Fig. 9. The changes of the semi-major axis (left) and inverse flattening (right) of the ’most up-to-date ellipsoid’ in the time. The first data indicates the geoid shape of Europe, then the colonial surveys altered these values, and finally the global values are provided.

The semi-major axis and the flattening of the estimated ellipsoids are not independent from each other. Fig. 9 shows the changes of these two figures as a function of the time, concerning the most accepted ellipsoids of that time, from 1800 to nowadays. The first part of this period was characterized by the increase of the semi-major axes and the decrease of the inverse flattening. The Earth occurred to be slightly larger and more sphere-like that it was first estimated. However, estimating the semi-major axis and the flattening is not a very complicated technical exercise. So, why were the results different, why is this whole change?

The first observers published the results based on just one arc measurement. The first ellipsoid, that was based on multiple, namely five, independent observations were set up by the Austrian scholar Walbeck in 1819. It occurred that the virtual semi-major axis and the flattening is changing from place to place. So, the whole body is not exactly an ellipsoid. It is almost that, but not completely.

This ’not completely’ occurred again during the building up the triangulation networks (see point 3.3). Because of this, the shape description based on the gravity theory, mentioned in the introduction of this chapter, was defined first by Karl Friedrich Gauss in the 1820s. The name ’geoid’ was proposed by Johann Benedict Listing much later, in 1872. Known the real shape of the geoid (Fig. 10) we can easily interpret the trend of the estimated ellipsoid parameters. Based on the European part of the geoid, the Earth seems to be smaller and more flattened. However, if we measure also in other continents, like in the locally different-shaped India, then we got the trend-line of the Fig. 9.

Fig. 10. The geoid, the level surface of the Earth, with massive vertical exaggeration.

The parameters of the most up-to-date ellipsoids, such as the GRS80, the WGS84, were determined by the whole geoid, with the following constraints:

• the geometric center of the ellipsoid should be at the mass center of the Earth

• the rotational axis of the ellipsoid should be at the rotational axis of the real Earth

• the volume of the ellipsoid and the geoid should be equal, and

• the altitude difference between the ellipsoid and the geoid should be on minimum, concerning the whole surface of the planet.

At a point of the surface, the geoid undulation is the distance of the chosen ellipsoid and the geoid along the plumb line. The geoid undulation from the best-fit WGS84 ellipsoid along the whole surface does not exceed the value of ±110 meters.

Summarizing: the equatorial radius of our planet is about 6378 kilometers, the difference between the equatorial and the polar radius (the error of the spherical model) is about 21 kilometers, while the maximum geoid undulation (the error of the ellipsoid model) is 110 meters.