3.3 Types of the triangulation networks, their set up and adjustment

Measuring of the distance of two points is possible by making a line between them and by placing a measuring rod along it – supposed the distance is not too long between our points. As the distance becomes longer, this procedure starts to be complicated and expensive: distances of more than a several hundred meters are very hard to measure this way. If the terrain between our points is rough or inpassable, this method cannot be applied at all.

Sketch of the Frisius triangulation

Fig. 11. The sketch of the Belgian triangulation of Gemma Frisius from the 16th century.

A new method was introduced at the end of the 16th, early 17th centuries. Measuring a longer distance can be made by measuring a shorter line and some angles. The first triangulation was proposed by Gemma Frisius (Fig. 11), then in 1615, another Dutchman, Snellius accomplished a distance measurement by triangulation between the towers of Alkmaar and Breda (true distance cca. 140 kilometers, throughout the Rhein-Maas delta swamps; Fig. 12). During the campaign, he set up triangles with church towers at the nodes and measured the angles of all triangles. Having these data, it was needed to measure only one triangle side to calculate all distances between the nodes.

Map of the Snellius triangulation

Fig. 12. The distance between the towns of Alkmaar (in the north) and Breda (in the south) was determined by the 1615 triangulation of Snellius, throughout swamps, marshlands and rivers.

The Snellius-measurements provided an interesting invention: the sum of the detected inner angles of a triangle occurred to be slightly more than 180 degrees (Fig. 13). This is the consequence of the non-planar, spheroid geometry of the surface of the Earth, and this is true at spherical triangles. It was the root of a new branch of the geometry: the spherical trigonometry.

Table of angles seen from a geodetic base point

Fig. 13. Angles between far geodetic points from the point of Johannes Berg, Budapest (Habsburg survey, 1901). The sum of the angles exceed the 360 degrees, according to the spheric surface.

Using triangulation networks, not only distances but also coordinates can be determined. For this, it is first needed to measure the geographic coordinates of one point of the network. That’s why there is in most cases an astronomical observatory at the starting point of a geodetic network: these measurements can be carried out there in most simple way. Also it is necessary a baseline: a shorter distance between two network points, whose distance is measured physically, and an azimuth: the measured angle between the true north and a triangle edge to a selected network point. Of course, the angles of all triangles should be measured together with the heights of the points. Using all of these data, and assuming an ellipsoid with a pre-set semi-major axis and flattening, the coordinates of all network points can be calculated. They are called triangulated coordinates. The longitude values in these coordinates are measured from the meridian of the astronomical observatory (Fig. 14).

Sketch of a historical geodetic network with triangles and baselines

Fig. 14. Sketch of the 1901 triangulation network between Vienna and Budapest.

To check the obtained coordinates of the base points, more baselines and – which later brought a real revolution in the data processing – the astronomical coordinates were observed at several triangulation points (at the so-called Laplace-points) of the network. The observed positions, however, differed from the ones, computed by the trigonometry. The difference occurred in all cases and its magnitude was not predictable. Its cause is the geoid shape of the Earth: the astronomic observations are based on the knowledge of the local horizontal and vertical lines, which are slightly different from the tangent and normal directions of the ellipsoid. As we mentioned above, the whole body is not exactly an ellipsoid. It is almost that, but not completely.

This problem became so important in the first half of the 19th century that Gauss invented his famous method of the least squares exactly to solve that. The goal is to ‘adjust’ the coordinates of the base points in order to minimize the squares of the differences occurring at the Laplace-points. The method is called geodetic network adjustment, which is, in practical words, to homogenize the errors, mostly caused by the geoid shape, in the whole network. The result of the adjustment is a geodetic point set organized into a network, with their finalized coordinate values.

What means the adjustment from geometric point of view? What is the geometric result? An ellipsoid whose

The geometric center of this ellipsoid is, of course, different from the mass center of the Earth (Fig. 15). This way, not only the size and shape of the ellipsoid is known but also its spatial location.

Fitting an ellipse to an irregular ellipse-like shape

Fig. 15. Geometric result of the geodetic network adjustment: fitting the ellipsoid to the surveyed part of the geoid; the geometric centre differs from the mass center of the Earth.

From the point of view of the ellipsoid location method in space, there are three types of them: