Information about triangulation

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

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, rad or 180 degrees.
The law of sines
The law of cosines
The Pythagorean theorem

Calculation

Triangulation

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 law of sines and law of cosines

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Now we can calculate AC and BC

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Last step is to calculate RC via

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MC can be calculated using the Pythagorean theorem

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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.

Mathematics and computer science

Subdivisions of spaces into triangles or higher dimensional simplices:

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Trigonometry (from Greek trigōnon "triangle" + metron "measure"^[1]), informally called trig, is a branch of mathematics that deals with triangles, particularly triangles in a plane where one angle of the triangle is 90 degrees (right angled
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Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. Geometry is one of the oldest sciences.
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coordinate system is a system for assigning an n-tuple of numbers or scalars to each point in an n-dimensional space. "Scalars" in many cases means real numbers, but, depending on context, can mean complex numbers or elements of some other commutative ring.
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A triangle is one of the basic shapes of geometry: a polygon with three corners or and three sides or edges which are straight line segments.

In Euclidean geometry any three non-collinear points determine a triangle and a unique plane, i.e.
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law of sines (or sine law, sine formula) is a statement about arbitrary triangles in the plane. If the sides of the triangle are a, b and c and the angles opposite to those sides are A, B and C
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trigonometric functions (also called circular functions) are functions of an angle. They are important in the study of triangles and modeling periodic phenomena, among many other applications.
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Euclidean geometry is a mathematical system attributed to the Greek mathematician Euclid of Alexandria. Euclid's text Elements is the earliest known systematic discussion of geometry.
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radian, in mathematics, is a unit of plane angle, equal to 180/π degrees, or about 57.2958 degrees. It is represented by the symbol "rad" or, more rarely, by the superscript c (for "circular measure"). For example, an angle of 1.2 radians would be written as "1.
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law of cosines (also known as Al-Kashi law or the cosine formula or cosine rule) is a statement about a general triangle which relates the lengths of its sides to the cosine of one of its angles. Using notation as in Fig.
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In mathematics, the Pythagorean theorem (AmE) or Pythagoras' theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. The theorem is named after the Greek mathematician Pythagoras, who by tradition is credited with its discovery and
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Surveying is the technique and science of accurately determining the terrestrial or three-dimensional space position of points and the distances and angles between them. These points are usually, but not exclusively, associated with positions on the surface of the Earth, and are
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Navigation is the process of planning, recording, and controlling the movement of a craft or vehicle from one place to another.^[1] The word navigate is derived from the Latin roots navis meaning "ship" and agere meaning "to move" or "to direct.
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worldwide view of the subject.
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Metrology (from Greek 'metron' (measure), and 'logos' (study of)) is the science of measurement.
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Astrometry is the branch of astronomy that relates to precise measurements and explanations of the positions and movements of stars and other celestial bodies. Although once thought of as an esoteric field with little useful application for the future, the information obtained by
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Binocular vision is vision in which both eyes are used together. The word binocular comes from two Latin roots, bin for two, and oculus for eye. Having two eyes confers at least four advantages over having one.
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Model rocketry is a hobby similar to building model airplanes, where rocket-shaped models are flown vertically and recovered by a variety of means (see Recovery below).
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weapon is a tool used to injure, incapacitate, or kill an adversary.^[1]^[2] Weapons may be used to attack and defend, and consequently also to threaten or protect. Metaphorically, anything used to damage (even psychologically) can be referred to as a weapon.
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A mesh is similar to fabric or a web in that it has many connected or weaved pieces. In clothing, a mesh is often defined as a loosely woven fabric that has a large number of closely-spaced holes, frequently used for modern sports jerseys and other clothing.
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In mathematics, simultaneous equations, or systems of equations, are a set of equations containing multiple variables. To solve simultaneous equations, the solver needs to use the provided equations to find the exact value of each variable.
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retriangulation of Great Britain, an immense task which involved erecting concrete triangulation pillars (trig points) on prominent hilltops throughout Great Britain.
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GSM localization is the use of multilateration to determine the location of GSM mobile phones, usually with the intent to locate the user.

Types of LBS Technology

Localization-Based Systems can be broadly divided into:

Network based
Handset based

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Multilateration, also known as hyperbolic positioning, is the process of locating an object by accurately computing the time difference of arrival (TDOA) of a signal emitted from the object to three or more receivers.
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Parallax, or more accurately motion parallax (Greek: παραλλαγή (parallagé) = alteration) is the change of angular position of two stationary points relative to each other as seen by an observer, caused by the motion of an
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Resection has multiple meanings:

Resectioning involves enlarging the cross-section of a river channel by deepening or widening the river to increase its hydraulic efficiency. This allows a larger discharge to be contained within the channel.
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SOCET SET is a software application that performs a variety of functions related to photogrammetry. It is developed and published by BAE Systems. SOCET SET is notable because it was the first commercial digital photogrammetry software program.
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trigonometrical station, triangulation pillar or trig point is a fixed surveying station for the geodetic surveying and other surveying projects on nearby areas.
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Triangulation

Calculation

See also

Mathematics and computer science

Types of LBS Technology