Triangulation

Triangulation determines the position of an inaccessible point by measuring the angles at which it is sighted from the two ends of a base of known length. The law of sines then gives the sought distances.

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How to measure it in class

FizziQ’s theodolite measures the angles needed for a triangulation: the elevation from the accelerometer, the azimuth from the magnetometer. All that remains is to measure a base on the ground.

Steps:

• Choose a clearly visible inaccessible target: a bell tower, a pylon, a tree on the other side of a river.
• Mark out a base AB on the ground, of a length comparable to the sighted distance, and measure it with a surveyor’s tape. This is the most critical measurement of the whole experiment.
• From point A, open the Theodolite tool and record the azimuth of the target C, then the azimuth of point B. The difference gives the angle  of the triangle.
• Repeat the operation from point B to obtain the angle B̂.
• Calculate Ĉ = 180° − Â − B̂, then apply the law of sines to obtain the distances AC and BC.
• Check the result with a second method, for example a map or a direct measurement when possible, and compare the deviations.

Scientific activities on this topic

• Triangulation with a theodolite - determine the distance of an inaccessible point from two sightings and a measured base.
• Law of sines with a theodolite - verify the law of sines experimentally on a triangle built in the field.
• Tree height with a theodolite - the simplest case of triangulation, in a vertical right triangle.

Learn more

The first uses of triangulation date back to antiquity, with notable examples in Egypt and Greece. However, the approach was largely developed and refined over time. It was the Dutch mathematician Willebrord Snell who, at the beginning of the 17th century, formalized triangulation as a geodetic method by measuring an arc of meridian in the Netherlands through a chain of triangles.

Modern triangulation was developed in the 18th century in France by geodesists such as César-François Cassini de Thury and Pierre Méchain, notably with the measurement of the length of the terrestrial meridian, an imaginary line from the North Pole to the South Pole. The project aimed to measure an arc of the terrestrial meridian to establish a universal unit of measurement, the meter. The Cassinis had already made preliminary measurements of the meridian between Paris and Dunkirk. However, Méchain joined the project in 1792 to measure the southern portion of the arc, from Rodez to Barcelona. The geodesists faced numerous challenges, including unfavorable weather conditions, instrument alignment problems, controversies with other geodesists and even calculation errors. Triangulation, an essential method used to measure distances and angles between points, was crucial but complex. Nevertheless, despite these difficulties, Méchain succeeded in completing a large portion of the meridian arc, thus contributing significantly to the French meridian. This major project laid the foundations of the modern metric system, which is still in international use today.

A triangulation measurement allows the determination of distances between inaccessible points using measured angles and known base lengths.

To perform a triangulation measurement using the law of sines, the following steps are used:

• Choosing triangulation points: Carefully select the triangulation points, ensuring they are visible from your main measurement point (total station) and that they form a triangle with the point you are trying to measure.
• Equipment setup: Install your measuring equipment, such as a total station, at a known reference point (your main measurement station) and align it with the triangulation points.
• Angle measurement: Use the equipment to measure the angles between the lines of sight from the total station to the triangulation points. You thus obtain the measurements of angles ∠A, ∠B and ∠C of the triangle.
• Base measurement: Measure the length of the base (BC in the example below) between two of the triangulation points using a laser rangefinder or a distance meter if the base is not known.
• Length calculation: Use the law of sines to calculate the lengths of the unknown sides of the triangle using the measured angles and the known base length: a/sin(A) = b/sin(B) = c/sin(C). For example, if you have measured ∠A and ∠B, and you know the length of the base BC, you can use the law of sines to calculate the length AC (or AB, or BC, depending on what you are trying to measure).
• Repetition: Repeat this process for each point you want to measure.
• Final calculation: Once you have measured all the necessary triangles, use the obtained lengths to calculate the final distance between your main measurement station and the point you are trying to measure.
• Correction: Do not forget to take into account the necessary corrections, such as the atmospheric refraction correction, to obtain accurate results.

History

One of the best-known applications of triangulation is the calculation of the length of the meridian by Pierre Méchain and Jean-Baptiste Joseph Delambre. These two French scientists were commissioned in 1792 by the Academy of Sciences to measure the length of the meridian, that is, the distance between the North Pole and the South Pole through Europe. To do this, they used triangulation to measure the angles between points located at known distances, then used these angles to determine the distance between these points. Méchain made his measurements in France and Spain, while Delambre made measurements in France. The two scientists concluded that the meridian was longer than previously thought, which made it possible to determine the shape and size of the Earth with greater precision.

Triangulation and trilateration: two methods not to be confused

Triangulation determines a position from angles measured from the ends of a known base. Trilateration, for its part, uses only distances. The distinction is far from academic: GPS is a trilateration system, not a triangulation system, contrary to what is often read. The receiver measures propagation times, hence distances to the satellites, and never angles. Historically, triangulation prevailed because measuring an angle with a theodolite was much easier than measuring a long distance; the arrival of laser rangefinders and atomic clocks has reversed this relationship.

Why a chain of triangles

One almost never triangulates with a single triangle. Geodesists covered the territory with a network of adjacent triangles, each resting on a side of the previous one. A single base was physically measured on the ground, with extreme care, and all the other lengths followed from it by calculation. The Melun base, measured by Delambre in 1798 over about 6 km using platinum rules, thus served as the reference for the entire French meridian.

The quality of the triangle determines the precision

A very flattened triangle, one of whose angles is close to 0° or 180°, gives poor results: a small angular error there produces a large position error. It is the same phenomenon as the dilution of precision in GPS. The practical rule is to choose triangles whose angles are all between 30° and 120°, and therefore to take a base of a length comparable to the distance to be measured. A short base facing a distant target is the main experimental design error.

Sources of error in class

Two dominate. First, the measurement of the base on the ground: a 20 m base paced out on irregular or sloping ground can easily be off by 1 m, that is 5 % which carries over entirely into the result. Second, the holding of the device: a smartphone held freehand, not exactly horizontal or not exactly aligned with the line of sight, introduces several degrees of error. Added to this, for the azimuth, is the magnetometer’s sensitivity to metal: stay away from fences, cars and concrete reinforcement.

Orders of magnitude

Angular error of a smartphone in azimuth: 2 to 5°. Typical precision of a school triangulation over 100 m: 5 to 15 %. Construction theodolite: 10 arc seconds. Typical side of the triangles of the French meridian: 30 to 40 km. Melun base measured in 1798: about 6 km, with an estimated uncertainty of a few centimeters. Stellar parallax of Proxima Centauri: 0.77 arc second, obtained by triangulation on a base equal to the diameter of the Earth’s orbit.

Formula

In a triangle ABC, the law of sines relates each side to the sine of the opposite angle:

a / sin(Â) = b / sin(B̂) = c / sin(Ĉ)

where:

• a, b, c: lengths of the sides opposite the vertices A, B, C respectively (m)
• Â, B̂, Ĉ: angles at the vertices (°), with  + B̂ + Ĉ = 180°

In practice, knowing the base c = AB and the two angles  and B̂ measured with the theodolite, the distance from point A to the target C is:

b = AC = c × sin(B̂) / sin(Ĉ) with Ĉ = 180° − Â − B̂

In the particular case of a right triangle, measurement of a height from a horizontal distance d:

h = d × tan(α)

where:

• h: height of the object above the sighting level (m)
• d: horizontal distance to the object (m)
• α: elevation angle (°)

When two distances and the angle between them are known, the law of cosines gives the third side:

a² = b² + c² − 2bc × cos(Â)

Application examples

• Base AB of 50 m, measured angles  = 68° and B̂ = 74°. Then Ĉ = 38°, and AC = 50 × sin(74°) / sin(38°) = 50 × 0.961 / 0.616 = 78 m.

• Measuring a tree: 20 m from its foot, top sighted at 35°. Height above eye level = 20 × tan(35°) = 14.0 m, to which 1.6 m of eye height is added, giving 15.6 m.

• The French meridian measured by Delambre and Méchain between 1792 and 1799 provided the length of the quarter of the terrestrial meridian, one ten-millionth of which defines the meter.

• Astronomers measure the distance of nearby stars by parallax: they sight the same star six months apart, using the diameter of the Earth’s orbit (300 million km) as a triangulation base.

• Locating an earthquake epicenter combines the two approaches: the distances deduced from the offset between P and S waves belong to trilateration, while networks of directional antennas triangulate the direction of arrival.

• An antenna mast, a pylon or a factory chimney can be measured in class by triangulation, without ever approaching them.

FAQ

Q: Does GPS work by triangulation? A: No, by trilateration. The receiver measures no angle: it measures the propagation time of the signals, hence its distance to each satellite, and looks for the point located at those distances. The word “triangulation” is used incorrectly in most popular accounts.

Q: What base length should be chosen? A: A base of the same order of magnitude as the distance to be measured. With a base that is too short, the two sightings are almost parallel, the triangle is very flattened and the slightest angular error makes the uncertainty on the distance explode.

Q: Why does my triangulation give an absurd result? A: First check that the sum of the measured angles is less than 180°: if  + B̂ exceeds 180°, one of the sightings is wrong. The most frequent cause is a magnetic disturbance of the azimuth by nearby metal.

Q: Can one triangulate with a single measurement point? A: No for a horizontal distance: two distinct viewpoints are needed, hence a base. On the other hand, for a height, the ground itself provides the base: the horizontal distance to the object plays this role and a single angle is enough.

Q: Should the curvature of the Earth be taken into account? A: Not in class. Over a few hundred meters, the correction is far smaller than the measurement uncertainty. It becomes indispensable for triangles of several tens of kilometers, where one works in spherical trigonometry and also corrects for atmospheric refraction.

Related concepts

Theodolite - Magnetometer - Accelerometer