02/10/2019

How trigonometry is behind GPS navigation

Imagem: Freepik

Reproduction of Marcelo Viana's column in Folha de S.Paulo.

At the end of the 18th century, France adopted the "meter" as its official unit of length, defined as 1/40,000,000 of the length of the Paris meridian. The problem is that it is impossible to measure a meridian directly. The solution was to choose two cities on the Paris meridian, Dunkirk and Barcelona, and measure the distance and the difference in latitude between them: from there, the length of the meridian could be obtained using a rule of three.

But the task remained complicated, as the distance between these cities is more than 1,000 km… Until the 1980s, distances between points on the Earth's surface – possibly separated by mountains, lakes, etc. – were calculated using the triangulation method, based on trigonometry.

The idea is as follows: we start with two points, A and B, such that the distance between them is known. Given another point, C, visible from both, we proceed as follows: at point A, we measure the angle between directions AB and AC, and at point B we measure the angle between directions AB and BC. This is done using a type of telescope called a theodolite. With this information, using trigonometric functions, it is possible to calculate the distances between A and C and between B and C. Then, we can calculate the distances from A and C (or B and C) to another point D, and so on.

This method allowed astronomers Jean-Baptiste Delambre (1749 – 1822) and Pierre Méchain (1744 – 1804) to accurately measure the distance from Dunkirk to Barcelona between 1792 and 1799. From these results, the first official definition of the meter was given.

But the use of trigonometry in cartography had begun earlier. In France, it was closely linked to the Cassini family, one of the most notable dynasties in the history of science. In the 1670s, the Astronomer Royal Giovanni Domenico Cassini (1625–1712) began a project to map all of France. Together with his son, Jacques Cassini (1677–1756), he completed in 1718 the first measurement of the distance from Dunkirk to Barcelona, which would be used to build provisional prototypes of the meter while awaiting Delambre and Méchain to finish their work.

Jacques' son, César-François Cassini (1714–1784), built upon the work of his father and grandfather to achieve the first complete triangulation of French territory. His son, Jean-Dominique Cassini (1748–1845)—great-grandson of Giovanni Domenico, who had founded the dynasty in the previous century—refined and completed his father's work. The Cassini map, published by the two between 1744 and 1793, established the standard for scientific cartography.

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