Trilateration 1

Trilateration

Figure 1. The plane, z=0, showing the 3 sphere centers, P1, P2, and P3; their x,ycoordinates; and the 3 sphere radii, r1, r2, and r3. The two intersections of the three

sphere surfaces are directly in front and directly behind the point designated intersectionsin the z=0 plane.

This article describes a method fordetermining the intersections of threesphere surfaces given the centers andradii of the three spheres.

More generally, trilateration methodsinvolve the determination of absoluteor relative locations of points bymeasurement of distances, using thegeometry of spheres or triangles.[1] [2]

[3] [4] In contrast to triangulation itdoes not involve the measurement ofangles.

Trilateration is mainly used insurveying and navigation, includingglobal positioning systems (GPS).

In a 2D plane using two referencepoints is normally sufficient to leaveonly two possibilities for the locationdetermined, and the tie must be brokenby 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 including afourth reference point or other information.

DerivationThe solution is found by formulating the equations for the three sphere surfaces and then solving the three equationsfor the three unknowns, x, y, and z. To simplify the calculations, the equations are formulated so that the centers ofthe spheres are on the z=0 plane. Also the formulation is such that one center is at the origin, and one other is on thex-axis. It is possible to formulate the equations in this manner since any three non-colinear points lie on a plane.After finding the solution it can be transformed back to the original three dimensional Cartesian coordinate system.We start with the equations for the three spheres:

We need to find a point located at (x, y, z) that satisfies all three equations.First we subtract the second equation from the first and solve for x:

.

We assume that the first two spheres intersect in more than one point, that is that d-r1 < r2 < d+r1. In this casesubstituting the equation for x back into the equation for the first sphere produces the equation for a circle, thesolution to the intersection of the first two spheres:

Trilateration 2

.

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 formula for the firstsphere 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 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 andintersecting that with the third sphere. If that circle falls entirely outside or inside of the sphere, z is equal to thesquare root of a negative number: no real solution exists. If that circle touches the sphere on exactly one point, z isequal to zero. If that circle touches the surface of the sphere at two points, then z is equal to plus or minus the squareroot of a positive number.

Preliminary and final computationsThe section Derivation pointed out that the coordinate system in which the sphere centers are designated must besuch that (1) all three centers are in the plane, Z = 0, (2) the sphere center, P1, is at the origin, and (3) the spherecenter, P2, is on the X axis. In general the problem will not be given in a form such that these requirement are met.This problem can be overcome as described below where the points, P1, P2, and P3 are treated as vectors from theorigin where indicated. P1, P2, and P3 are of course expressed in the original coordinate system.

is the unit vector in the direction from P1 to P2.

is the signed magnitude of the x component, in the figure 1 coordinate system, of thevector from P1 to P3.

is the unitized y component, in the figure 1 coordinate system, of the vector

from P1 to P3. Note that : has been defined in such a manner that the points P1, P2, and P3 are all in thez=0 plane of the figure 1 coordinate system as required.

The third basis unit vector is . Therefore,the distance between the centers P1 and P2 and

is the signed magnitude of the y component, in the figure 1 coordinate system, of thevector from P1 to P3.

Using as computed above, solve for x, y and z as described in the Derivation section. Then

gives the points in the original coordinate system since , the basis unit vectors, are expressed inthe original coordinate system.

Trilateration 3

ApplicationTrilateration 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 or more

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. html)[4] free dictionary (http:/ / www. thefreedictionary. com/ trilateration)

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Article Sources and ContributorsTrilateration  Source: http://en.wikipedia.org/w/index.php?oldid=392188782  Contributors: 4johnny, AMag, Adrian.benko, Ahseaton, Antandrus, Ashawley, Barrylb, Bepa, Braindrain0000,Brews ohare, Bubblebob, Charles Matthews, Crazy Software Productions, Dcoetzee, Denelson83, Discospinster, Dreadstar, Dtcdthingy, Frelke, Friesse, Fyyer, GCW50, Giftlite, Interrested,Jen.carrol, Jleedev, Junkyardprince, Kwamikagami, Leonard G., Limit i, Linas, Lou.weird, Lupin, Maximus Rex, Mercenario97, Mfolozi, Mhaitham.shammaa, Michael Hardy, Mikewax,MrOllie, Nbarth, Nk, Noideta, Nominal animal, Oleg Alexandrov, PEHowland, RHB100, Robert Weemeyer, Robofish, Rossi, Sicooke, Siddhant, Superdude99, The Anome, Tresiden, Wirelessfriend, Wonglijie, Woodstone, ZooFari, 86 anonymous edits

Image Sources, Licenses and ContributorsImage:3spheres.jpg  Source: http://en.wikipedia.org/w/index.php?title=File:3spheres.jpg  License: Free Art License  Contributors: User:Rhb100

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