TrilaterationLinear Equations > Applications > Trilateration
Denote by , the three known points and by the measured distances to the emitter. Mathematically the problem is to solve, for a point , the equations , . We write them out: Let . The equations above imply that Using matrix notation, with the matrix of points, and the vector of ones: Let us assume that the square matrix is full-rank, that is, invertible. The equation above implies that In words: the point lies in a line passing through and with direction . We can then solve the equation in : . This equation is quadratic in : and can be solved in closed-form. The spheres intersect if and only if there is a real, non-negative solution . Generically, if the spheres have a non-empty intersection, there are two positive solutions, hence two points in the intersection. This is understandable geometrically: the intersection of two spheres is a circle, and intersecting a circle with a third sphere produces two points. The line joining the two points is the line , as identified above. |