# Trilateration and the Intersection of Three Spheres

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Trilateration is used in technologies such as GPS to find the exact location of a point on Earth or in space. It determines a location by means of three distances to known points in space, such as orbiting satellites.

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Contributed by: Erik Mahieu (December 2015)

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

All three centers of the spheres are in the plane : at , at , at .

,

,

.

The two intersection points are found by solving these three equations for , , and .

The intersection circles have the parametric form of a circle on a sphere, centered at the origin and with the axis as its normal, rotated by an angle around the axis and around the axis:

,

,

,

where

is the angular radius of the circle,

is the rotation around the axis, is the rotation around the axis,

and is a parameter running around the circle, from 0 to .

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