Trilateration

Remarks

Example: $p_1 = (0,0,0)$, $p_2 = (2,0,0)$, $p_3 = (0,2,0)$ and $r_1 = r_2 = r_3 = \sqrt{3}$ give $x = (1, 1, \pm 1)$; indeed $\norm{(1,1,1) - (2,0,0)}^2 = 1 + 1 + 1 = 3$. Non-example: with only $p_1$ and $p_2$, the solutions form a circle, so two distances do not determine $x$. Subtracting the first equation from the others gives linear equations $2\ip{x}{p_1 - p_i} = r_i^2 - r_1^2 - \norm{p_i}^2 + \norm{p_1}^2$. If $p_1, p_2, p_3$ are not collinear, there are at most two solutions, mirror images in the plane through $p_1, p_2, p_3$. If $n \ge 4$ and the $p_i$ are not coplanar, the solution is unique when it exists. With measurement errors, $x$ is found by least squares.