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
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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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