Liverpool F.C.

Mathematics

In Topology, a triangulation of a topological Espace X is a Complexe simplicial K homeomorphic with X , and a Homéomorphisme H : K → X . In the terms of Layman, if X is a plan then a triangulation is a way of cutting out X in a collection of Triangle S.

The triangulation is useful to determine the properties of a topological space.

A traditional example is the Triangulation of Delaunay.

The triangulation is also the process which makes it possible to determine a distance by calculating the length of the one on the sides of a triangle, and by measuring two angles of this triangle. This method uses trigonometrical identities.

Six hundred years before the Christian era, Thalès developed a method to at sea evaluate the distance from a boat at the coast. To have an approximate measurement of this distance, it placed two observers has and C on the shore distant from a known distance B . He asked each one of them to measure the angle which the lines form passing by the boat B and one of them, and line passing by the two observers:

The method has an interest if we want to determine long distances; but in this case we must place the two observers sufficiently distant one from the other, so that measurements of angle are more precise.

The properties often used for the triangulation are:

the sum of the angles of a triangle is equal to π Radian S (180 degree S).
the Law of the sines
the Theorem of Al-Kashi
the Theorem of Pythagore

Triangulation of a country

Until in the Years 1980, one used primarily the triangulation to measure the distances. If one wants to measure the distance between two points B and C , the operation consists in taking a point of reference has , then to measure the angles which form the lines ( AB ) and ( AC ) compared to North, which gives the angle ( BÂC ).

This process, repeated gradually, was used by Delambre and Méchain of 1792 with 1798 to measure the distance between Dunkirk and Barcelona (approximately: 1147 km) on the Méridien of Paris, which will allow the first practical and official definition Mètre in 1799 (although the design of the meter itself as a universal unit and a decimal is quite former, cf work of John Wilkins and Tito Livio Burattini).

Starting from a point of reference, one can thus determine the position of the various points of a territory and carry out a grid. This grid then makes it possible to have a precise Cartographie and whose deformations are known, compared to the charts which were drawn by a show of hands starting from a high point. The first chart of France thus traced was published in 1745, starting from the statements of Jacques and César Cassini.

Applications

Triangulation by statement of the directions

The triangulation is used in various sectors, like the Survie, the Navigation, the Astronomie, in the armament (fused).

A ship can thus know its position by raising the direction of observation (angle compared to North) of two distant points (for example a bell-tower of church, a headlight); it is enough for him then, on a chart, to plot the straight lines passing by the points observed and having the raised direction, the intersection of these lines being the position of the ship. To raise the inaccuracies of measurement, one generally uses 3 benchmarks, called Amer S. It is the Navigation by raisings.

In the case of electromagnetic waves (for example of the waves radio), the position can be determined with a directional antenna (i.e. an antenna collecting only the waves coming from a given direction); the orientation for which the signal is strongest gives the direction of the transmitter, it is then enough to make several statements to have the position of the transmitter (Radiogoniométrie). This method for example was used during the German Occupation of France to detect the pirate radio station transmitters.

Static position

Two points are signed, and one raises the directions of aiming. It is then enough to plot, on a chart, a straight line passing by the point concerned and having the raised direction. The intersection of the right-hand sides gives the position.

There are two tops of the triangle (land-marks) and the direction on the two sides not uniting these tops (raisings), which makes it possible to determine the triangle completely.

If three readings are taken, one should obtain a single point of competition of the three lines. In practice, the inaccuracies - on the aiming, the reading of the angle, the layout of the right-hand side - make that one will obtain a triangle, the dimension of the triangle giving an estimate of the measuring accuracy. One can reasonably take for position the Barycentre of this triangle, and for error the distance between this center and the most distant point.

Moving traffic

In the case of a moving traffic, it is necessary to take into account the displacement of the vehicle. It is necessary for that to know the direction and the speed of the vehicle. The direction of the movement is given by the compass; in the case of a sailing ship, speed can be estimated starting from the speed of the wind and the current.

If speed is slow and that the statements are made in a close way (case of sea transport), one can neglect this phenomenon, on the other hand, it is necessary to note the hour of the statement.

The knowledge of this movement makes it possible to make a statement with only one land-mark, for example in the case of a navigation per time of fog where only a characteristic place would be visible intermittently. One then raises the directions and the hours of the statement.

There is thus a top of the triangle (the land-mark), the directions on two sides (two statements), and the direction and the length on the third side (trajectory of the boat), which makes it possible to determine the triangle completely.

See the article Navigation by raisings > Positioning with only one land-mark

Triangulation by statement of the distances

One can raise a position by estimating the distance compared to three points. If two points of reference are taken, there are two tops and the lengths on the sides not uniting these tops, which defines two triangles; the point of the statement is at the third top of the one of these triangle. The addition of a third point of reference makes it possible to determine which of the two tops is the good.

Use of the intensity of a signal

With an electromagnetic wave, one can use the intensity of the signal collected by a nondirectional antenna. If the propagation medium is homogeneous and isotropic, the intensity decreases according to the square of the distance (energy is distributed on a growing sphere). The intensity thus makes it possible to estimate the distance, and thus to locate the transmitter on a circle centered on the receiver. A second receiver makes it possible to trace a second circle, the transmitter is thus with the intersection of the two circles; a third receiver makes it possible to determine which of the two points of intersection is the good (or logic, a ship cannot be on the grounds). This method is also used in Sismologie to know the position of the epicentre of a seism; one considers the intersection of spheres then (one does not have more the constraint of a transmitter located on the surface), and it is necessary to correct calculations of the heterogeneity of the medium (variation of the index of refraction according to the depth, reflection and refraction on the coat…).

If one does not know the intensity of the transmitter nor the output of the receivers, it is necessary to be the positions for which the signal is received with the same intensity; the transmitter is then on the Médiatrice segment consisted the two receivers. One can locate the source with a second statement, with the intersection of the two mediating ones. The method of mediating, slightly modified, is used in Rescue-clearing to locate victims buried with a Géostéréophone.

Use of the propagation velocity of a signal

If there are events emitting a signal, then by noting the shift in the arrival of the signals, one can determine the difference in distance between the events and the receivers, on the condition of knowing the propagation velocity of the signal.

If the receiver itself is synchronized with the transmitters, one can then directly determine the time of way and thus the distance between transmitters and receiver.

For example, to locate a Seism, there is a single event generating the signal, the seism, and several synchronized stations receiving, the Sismographe S. the seismic Onde arrives at different times at the seismographs. The hearth of the seism (Hypocentre) is at a distance d_i seismograph I .

The hearth and two seismographs form a triangle; one knows two tops S ₁ and S ₂ of the triangle (seismographs), coordinates ( X ₁, there ₁, Z ₁) and ( X ₂, there ₂, Z ₂), and the difference in length D between the two sides not uniting these tops. The third top of the triangle is thus on a surface checking

d (M, S_1) - D (M, S_2) = D

that is to say

\ sqrt {(x-x_1) ^2 + (there there _1) ^2 + (z-z_1) ^2} - \ sqrt {(x-x_2) ^2 + (there there _2) ^2 + (z-z_2) ^2} = D

who is the equation of a Hyperboloïde. One needs sufficient seismographs to define three surface, the point of competition of these surfaces giving the hearth of the seism.

One has several types of seismic waves travelling at different speeds, which makes it possible to make several determinations. However, the problem is made complex because:

the waves do not travel in straight line: the propagation velocity thus depends on the pressure the depth under the ground, one has a phenomenon of Réfraction with continuous variation of index;
there are discontinuities, the waves thus undergo reflection S;
the waves of Cisaillement (standard S) do not travel in the liquid, therefore cannot pass by the terrestrial core.

See the article Measurement in seismology.

One can also locate other events such as for example a strong explosion (Nuclear test, industrial Catastrophe of great width like the Catastrophe of factory AZF…).

For navigation by satellite (standard system GPS), there are several synchronized transmitters and a single receiver at the place to be located.

See too

Resolution of a geodetic triangle
Arc of Struve
History of the Triangulation in France

External bonds

the triangulation and measures it distances, Observatoire of Paris

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