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Trilateration
Trilateration This article describes a method for determining the intersections of three sphere surfaces given the centers and radii of the three spheres. More generally, trilateration methods involve the determination of absolute or relative locations of points by measurement of distances, using the [1] [2]
geometry of spheres or triangles. [3] [4]
In contrast to triangulation it
does not involve the measurement of angles. Trilateration
is
mainly
used
in
surveying and navigation, including global positioning systems (GPS). In a 2D plane using two reference points is normally sufficient to leave
Figure 1. The plane, z=0, showing the 3 sphere centers, P1, P2, and P3; their coordinates; and the 3 sphere radii, r1, r2, and r3 . The two intersections of the
sphere surfaces are directly in front and directly behind the point designated inters in the z=0 plane.
only two possibilities for the location determined, and the tie must be broken by including a third reference point or other information.
In 3D space, using three reference points similarly leaves only two possibilities, and the tie is broken by inc fourth reference point or other information.
Derivation
The solution is found by formulating the equations for the three sphere surfaces and then solving the three eq
for the three unknowns, x, y, and z. To simplify the calculations, the equations are formulated so that the ce the spheres are on the z=0 plane. Also the formulation is such that one center is at the origin, and one other
x-axis. It is possible to formulate the equations in this manner since any three non-colinear points lie on After finding the solution it can be transformed back to the original three Sign dimensional up to vote onCartesian this title coordinate sy We start with the equations for the three spheres:
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Trilateration
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Substituting :
into the formula for the third sphere and solving for y there results: .
Now that we have the x- and y-coordinates of the solution point, we can simply rearrange the for mula for sphere to find the z-coordinate:
Now we have the solution to all three points x, y and z. Because z is expressed as the positive or negative squa 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 f rom intersecting the first and second sph
intersecting that with the third sphere. If that circle falls entirely outside or inside of the sphere, z is equa
square root of a negative number: no real solution exists. If that circle touches the sphere on exactly one po
equal to zero. If that circle touches the surface of the sphere at two points, then z is equal to plus or minus the root of a positive number.
Preliminary and final computations The section Derivation pointed out that the coordinate system in which the sphere centers are designated
such that (1) all three centers are in the plane, Z = 0, (2) the sphere center, P1, is at the origin, and (3) the
center, P2, is on the X axis. In general the problem will not be given in a form such that these requirement are
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This problem can be overcome as described below where the points, P1, P2, and P3 are treated as vectors f Unlock fullexpressed access withina the freeoriginal trial. origin where indicated. P1, P2, and P3 are of course coordinate system.
is the unit vector in the direction to P2. Download Withfrom FreeP1Trial
is the signed magnitude of the x component, in the figure 1 coordinate system vector from P1 to P3. is the unitized y component, in the figure 1 coordinate system, of from P1 to P3. Note that :
has been defined in such a manner that the points P1, P2, and P3 are
z=0 plane of the figure 1 coordinate system as required. The third basis unit vector is . Therefore,
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is the signed magnitude of the y component, in the figure 1 coordinate system
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Trilateration
Application Trilateration is mainly used in surveying and navigation, including global positioning systems (GPS).
See also • Euclidean distance • Multilateration - position estimation using measurements of time difference of arrival at (or from) three o sites. • Resection • Triangulation • Global positioning system
References [1] Encyclopedia Brittannica (http:/ / www.britannica.com/ EBchecked/ topic/ 605329/ trilateration) [2] diracdelta (http:/ / www.diracdelta.co.uk/ science/ source/ t/ r/ trilateration/ source.html)
[3] global maritime (http:/ / www.globmaritime.com/ 200902052139/ marine-navigation/ trilateration-traverse-and-vertical-surveying.htm [4] free dictionary (http:/ / www.thefreedictionary.com/ trilateration)
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Article Sources and Contributors
Article Sources and Contributors
Trilateration Source: http://en.wikipedia.org/w/index.php?oldid=392188782 Contributors: 4johnny, AMag, Adrian.benko, Ahseaton, Antandrus, Ashawley, Barrylb, Bepa, Braindr Brews ohare, Bubblebob, Charles Matthews, Crazy Software Productions, Dcoetzee, Denelson83, Discospinster, Dreadstar, Dtcdthingy, Frelke, Friesse, Fyyer, GCW50, Giftlite, Inter Jen.carrol, Jleedev, Junkyardprince, Kwamikagami, Leonard G., Limit i, Linas, Lou.weird, Lupin, Maximus Rex, Mercenario97, Mfolozi, Mhaitham.shammaa, Michael Hardy, Mikew MrOllie, Nbarth, Nk, Noideta, Nominal animal, Oleg Alexandrov, PEHowland, RHB100, Robert Weemeyer, Robofish, Rossi, Sicooke, Siddhant, Superdude99, The Anome, Tresiden friend, Wonglijie, Woodstone, ZooFari, 86 anonymous edits
Image Sources, Licenses and Contributors Image:3spheres.jpg Source: http://en.wikipedia.org/w/index.php?title=File:3spheres.jpg License: Free Art License Contributors: User:Rhb100
License Creative Commons Attribution-Share Alike 3.0 Unported http:/ / creativecommons.org/ licenses/ by-sa/ 3.0/
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