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Fig. 3. The position of the three sensors and the intersection of three spheres related to the
sensors
Now, the obtained value x is put in the one of the equations (27). We select the first one:
R 2
x 0 2
R 1
y 2
z 2
R 1
+
+
=
4 x 0
R 2
x 0 2
R 1
y 2
z 2
R 1
+
=
.
(29)
4 x 0
For convenience, the right side of the second equality of (29) is showed by R cir .
Thus, the
intersection of the two spheres is a circle with the following equation:
y 2
z 2
R cir ,
+
=
(30)
R 2
R 1
4 x 0
=
Which is located in the plane x
.
Then the intersection of this circle and the third sphere should be obtained. The third sphere
has the center C and radius R 3 . So its equation is:
2
x 2
z 2
R 3 .
+ (
y
y 0
)
+
=
(31)
The left side of the equation (31) is extended, and the circle equation is used in it:
x 2
2
z 2
y 0 +
x 2
y 2
z 2
+ (
y
y 0
)
+
=
2 yy 0
+
+
+
R 2
2
R 1
y 0 +
R cir
=
2 yy 0 +
+
4 x 0
R 2 R 1
4 x 0
2
y 0 +
R cir
+
y
=
.
(32)
2 y 0
 
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