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The application of trigonometry
for positioning
for positioning
Maps help us with orientating ourselves and getting to know our closer and wider environment. Creating them precisely and realistically is not an easy task. Mathematics, and actually trigonometry, played a major role in solving such problems.
Triangulation
The first method still applied today, which can be considered as scientific, is attached to the name of WILLEBRORD SNELLIUS.
The idea behind the method is that features are marked out in the area to be mapped so that some neighbouring points can be well seen from any marked point. These points define a triangular net.
The triangulation network is the set of points marked with concrete pillars or stones located with a uniform density all over the country. The points are located so that there is a direct line of sight from each one to the neighbouring points, as it is the condition of commensuration. In order to be able to see as far as possible the triangulation points were located on mountaintops and hilltops, or if there were no such places, then these were built on the church towers. The work was done with theodolites, later with \latex{ EDM } (Electronic Distance Measurement) equipment. Once the measurements were done (which lasted for several years, even decades) the network was equalised, and each point was given a coordinate in a unified system. To increase precision the least dense possible network was built (the first-order network in Hungary has a dot density of \latex{30\,km} or so), and when it was done the densification was started. The most dense state network of Hungary is fourth-order, and with the \latex{1.5}–\latex{2\,km} dot density it is creditable even internationally. So when a surveyor has to do a measurement, there is at least one point available within a radius of \latex{1\,km}, which has a coordinate in the unified national system.
A point is marked out \latex{\left(A\right)}, which is going to be the base point, and its coordinates are known. It is the same as if we knew the latitude and longitude on the map. We need to know the distance of at least one point measured from this point \latex{\left(AB\right)}. After that it will be enough to know the angle we see the other points at with reference to the line segment \latex{AB}. (Figure 53)
When starting from the line segment \latex{AB} we can get to any line segment of the triangular net in finitely many steps, and the distance of any two points of it can be determined with the help of the sine rule:
\latex{AC=AB\times \frac{sin \beta}{\sin\left(180°-\alpha-\beta\right)}=AB \times \frac{\sin \beta}{\sin \left(\alpha + \beta\right)}};
\latex{BC=AB\times \frac{sin \alpha}{sin (\alpha + \beta)}.}
Obviously the question is more complicated since in general these points are located at distinct heights. However these can be determined if the height above sea level is known for the base point. We can measure the angle of elevation and the angle of depression of the measured points with reference to this that makes it possible to determine the difference in levels compared to the base point.
\latex{A}
\latex{B}
\latex{C}
\latex{\alpha}
\latex{\beta}
\latex{\gamma}
Figure 53
In reality these calculations are much more complicated, since in the case of larger areas even the curved surface of the Earth needs to be taken into account, and obviously the errors made during the measurements need correction.
GPS (Global Positioning System)
This system uses the advantages resulting from the development of physics and technology, thus a much better precision can be reached compared to the previous measuring methods.
\latex{24} satellites orbit around the Earth at an altitude of about \latex{20,000\,km}. Their orbits are distributed so that at least \latex{4} satellites are always visible from any point on the Earth at any given instant. There are at least \latex{8} GPS satellites above Europe all the time. An observer system on the Earth constantly keeps a track of the precise cosmic position of the satellites. There is an atomic clock on each satellite. The atomic clocks continuously radiate the exact time according to their atomic clocks as radio signals.
In a GPS receiver there is a radio receiver and also a very precise clock. The receiver receives the precise time signals from the satellites, and compares them to the time of its internal clock. Time is needed for the radio waves to travel, therefore there is some difference between the two times. If we multiply this difference by the speed of light, we will get the distance between the given satellite and the observer.
If we perform this distance measurement for three satellites, we will know how far away we are from three points of the space. The intersection point of the surface of the spheres “drawn” around these points with the measured distances as radii will mark out our position. So we can give our position even with a precision of \latex{1}-\latex{2\,m}.
The GPS systems are widely used in air-traffic control, in sea navigation, in the determination of position of vehicles, in agriculture, in forestry, in cartography and in the control of robots. However, the military applications also have priority in this field.
GPS technology can do even more. For example the very slow tectonic movements can also be detected with it. Tracing these movements helps us with getting to know the processes of the interior of Earth, and it even serves as a realistic possibility to better predict earthquakes. For this it is obviously necessary to perform the measurements continuously, not only from time to time. The collection of the stations equipped with GPS receivers working all the time – the findings of which will be transferred to common, in most of the cases international processing centres – are called an active GPS network. Such a network is maintained by the International Association of Geodesy (IAG).
