Information about Trilateration
Trilateration is a method of determining the relative positions of objects using the geometry of triangles in a similar fashion as triangulation. Unlike triangulation, which uses angle measurements (together with at least one known distance) to calculate the subject's location, trilateration uses the known locations of two or more reference points, and the measured distance between the subject and each reference point. To accurately and uniquely determine the relative location of a point on a 2D plane using trilateration alone, generally at least 3 reference points are needed.
Starting with three spheres,
and
we subtract the second from the first and solve for x:
Substituting this back into the formula for the first sphere produces the formula for a circle, the solution to the intersection of the first two spheres:
Setting this formula equal to the formula for the third sphere finds:
Now that we have the x- and y-coordinates of the solution point, we can simply rearrange the formula for the first sphere to find the z-coordinate:
Now we have the solution to all three points x, y and z. Because z is expressed as a square root, it is possible for there to be zero, one or two solutions to the problem.
This last part can be visualized as taking the circle found from intersecting the first and second sphere and intersecting that with the third sphere. If that circle falls entirely outside of the sphere, z is equal to the square root of a negative number: no real solution exists. If that circle touches the sphere on exactly one point, z is equal to zero. If that circle touches the surface of the sphere at two points, then z is equal to plus or minus the square root of a positive number.
In the case of no solution, a not uncommon one when using noisy data, the nearest solution is zero. One should be careful, though, to do a sanity check and assume zero only when the error is appropriately small.
In the case of two solutions, some technique must be used to disambiguate between the two. This can be done mathematically, by using a fourth sphere with its center not being located on the same plane as the centers of the other three, and determining which point lies closest to the surface of this sphere. Or it can be done logically—for example, GPS receivers assume that the point that lies inside the orbit of the satellites is the correct one when faced with this ambiguity, because it is generally safe to assume that the user is never in space, outside the satellites' orbits.
This new model emphasizes the importance of choosing three points that are in very different directions — if the points are relatively close together and all far from the point being located, it will take very precise measurement to find the point using trilateration.
Derivation
A mathematical derivation for the solution of a three-dimensional trilateration problem can be found by taking the formulae for three spheres and setting them equal to each other. To do this, we must apply three constraints to the centers of these spheres; all three must be on the z=0 plane, one must be on the origin, and one other must be on the x-axis. It is, however, possible to translate any set of three points to comply with these constraints, find the solution point, and then reverse the translation to find the solution point in the original coordinate system.Starting with three spheres,
,
and
,
we subtract the second from the first and solve for x:
.
Substituting this back into the formula for the first sphere produces the formula for a circle, the solution to the intersection of the first two spheres:
.
Setting this formula equal to the formula for the third sphere finds:
.
Now that we have the x- and y-coordinates of the solution point, we can simply rearrange the formula for the first sphere to find the z-coordinate:
Now we have the solution to all three points x, y and z. Because z is expressed as a square root, it is possible for there to be zero, one or two solutions to the problem.
This last part can be visualized as taking the circle found from intersecting the first and second sphere and intersecting that with the third sphere. If that circle falls entirely outside of the sphere, z is equal to the square root of a negative number: no real solution exists. If that circle touches the sphere on exactly one point, z is equal to zero. If that circle touches the surface of the sphere at two points, then z is equal to plus or minus the square root of a positive number.
In the case of no solution, a not uncommon one when using noisy data, the nearest solution is zero. One should be careful, though, to do a sanity check and assume zero only when the error is appropriately small.
In the case of two solutions, some technique must be used to disambiguate between the two. This can be done mathematically, by using a fourth sphere with its center not being located on the same plane as the centers of the other three, and determining which point lies closest to the surface of this sphere. Or it can be done logically—for example, GPS receivers assume that the point that lies inside the orbit of the satellites is the correct one when faced with this ambiguity, because it is generally safe to assume that the user is never in space, outside the satellites' orbits.
Error model
When measurement error is introduced into the picture, things become a little more complicated. If we know that the distance from P to a reference point lies in a range (a closed interval) [r1, r2], then we know that P lies in a circular band between the circles of those two radii. If we know a range for another point, we can take the intersection, which will be either one or two areas bounded by circular arcs. A third point will usually narrow it down to a single area, but this area may still be of significant size; additional reference points can help shrink it further, but as the area shrinks more measurements quickly become less useful. In three dimensions, we are instead intersecting spherical shells with thickness, similar to bowling balls.This new model emphasizes the importance of choosing three points that are in very different directions — if the points are relatively close together and all far from the point being located, it will take very precise measurement to find the point using trilateration.
See also
- Multilateration - position estimation using measurements of time difference of arrival at (or from) three or more sites.
- Resection
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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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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In Euclidean geometry any three non-collinear points determine a triangle and a unique plane, i.e.
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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.
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angle (in full, plane angle) is the figure formed by two rays sharing a common endpoint, called the vertex of the angle. The magnitude of the angle is the "amount of rotation" that separates the two rays, and can be measured by considering the length of circular arc swept
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Distance is a numerical description of how far apart objects are at any given moment in time. In physics or everyday discussion, distance may refer to a physical length, a period of time, or an estimation based on other criteria (e.g. "two counties over").
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A 2D geometric model is a geometric model of an object as two-dimensional figure, usually on the Euclidean or Cartesian plane.
Even though all material objects are three-dimensional, a 2D geometric model is often adequate for certain flat objects, such as paper cut-outs and
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Even though all material objects are three-dimensional, a 2D geometric model is often adequate for certain flat objects, such as paper cut-outs and
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translation is moving every point a constant distance in a specified direction. It is one of the rigid motions (other rigid motions include rotation and reflection). A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin
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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 sanity test or sanity check is a basic test to quickly evaluate the validity of a claim or calculation. In mathematics, for example, when multiplying by three or nine, verifying that the sum of the digits of the result is a multiple of 3 or 9 respectively is a sanity test.
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Global Positioning System (GPS) is the only fully functional Global Navigation Satellite System (GNSS). Utilizing a constellation of at least 24 medium Earth orbit satellites that transmit precise microwave signals, the system enables a GPS receiver to determine its
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In algebra, an interval is a set that contains every real number between two indicated numbers and may contain the two numbers themselves. Interval notation is the notation in which permitted values for a variable are expressed as ranging over a certain interval; "" is an
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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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