Triangulation

This article is about measurement by the use of triangles. For other uses, see Triangulation (disambiguation).

Triangulation can be used to find the coordinates and sometimes distance from the shore to the ship. The observer at A measures the angle α between the shore and the ship, and the observer at B does likewise for β . If the length l or the coordinates of A and B are known, then the law of sines can be applied to find the coordinates of the ship at C and the distance d.

In trigonometry and geometry, triangulation is the process of finding coordinates and distance to a point by calculating the length of one side of a triangle, given measurements of angles and sides of the triangle formed by that point and two other known reference points, using the law of sines.

In the figure at right, the third angle of the triangle (call it θ) is known to be 180 − α − β, since the sum of the three angles in any triangle is known to be 180 degrees. The opposite-side for this (the third) angle is l, which is a known distance. Since, by the law of sines, the ratio sin(θ)/l is equal to that same ratio for the other two angles α and β, the lengths of any of the remaining two sides can be computed by algebra. Given either of these lengths, sine and cosine can be used to calculate the offsets in both the north/south and east/west axes from the corresponding observation point to the unknown point, thereby giving its final coordinates.

Some identities often used (valid only in flat or euclidean geometry):

• The sum of the angles of a triangle is π radians or 180 degrees.
• The law of sines
• The law of cosines
• The Pythagorean theorem

Calculation

• α, β and distance AB are already known
• C can be calculated by using the distance RC or MC:

• RC: Position of C can be calculated using the Law of Sines

$\backslash gamma=180^\backslash circ-\backslash alpha-\backslash beta$

$\backslash frac\{\backslash sin\backslash alpha\}\{BC\}=\backslash frac\{\backslash sin\backslash beta\}\{AC\}=\backslash frac\{\backslash sin\backslash gamma\}\{AB\}$

Now we can calculate AC and BC

$AC=\backslash frac\{AB\backslash cdot\backslash sin\backslash beta\}\{\backslash sin\backslash gamma\}$

$BC=\backslash frac\{AB\backslash cdot\backslash sin\backslash alpha\}\{\backslash sin\backslash gamma\}$

Last step is to calculate RC via

$RC=AC\; \backslash cdot\; \backslash sin\backslash alpha$

or

$RC=BC\; \backslash cdot\; \backslash sin\backslash beta$

• MC can be calculated using the Law of Cosines and the Pythagorean theorem

$MR=AM-RB=\backslash left(\backslash frac\{AB\}\{2\}\backslash right)-\backslash left(BC\; \backslash cdot\; \backslash cos\backslash beta\backslash right)$

$MC=\backslash sqrt\{MR^2+RC^2\}$

Triangulation is used for many purposes, including surveying, navigation, metrology, astrometry, binocular vision, model rocketry and gun direction of weapons.

Many of these surveying problems involve the solution of large meshes of triangles, with hundreds or even thousands of observations. Complex triangulation problems involving real-world observations with errors require the solution of large systems of simultaneous equations to generate solutions.

Famous uses of triangulation have included the retriangulation of Great Britain.

See also

• GSM localization
• Multilateration, where a point is calculated using the time-difference-of-arrival between other known points
• Parallax
• Resection
• SOCET SET
• Trig point
• Trilateration, where a point is calculated given its distances from other known points

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