Trilaterasi

Trilaterasi is a method of determining the position of a point by using the distances to three other known points. It is a fundamental concept in surveying and navigation. The principle relies on the geometric property that a point is uniquely defined by its distance from three non-collinear points.

In a two-dimensional plane, if the distances from an unknown point to three known points are d1, d2, and d3, and the coordinates of the known points are (x1, y1), (x2, y2), and (x3, y3), then the unknown point (x, y) can be found by solving the system of equations:

(x - x1)^2 + (y - y1)^2 = d1^2

(x - x2)^2 + (y - y2)^2 = d2^2

(x - x3)^2 + (y - y3)^2 = d3^2

Each equation represents a circle centered at one of the known points with a radius equal to the measured distance. The intersection of these three circles will ideally yield a single point, which is the location of the unknown point. In practice, due to measurement errors, the circles may not intersect at a single point, and a best-fit solution is typically sought.

In three-dimensional space, trilaterasi requires distances to at least four known points to uniquely determine the position in x, y, and z coordinates. Each distance measurement defines a sphere, and the intersection of these spheres pinpoints the location. This method is widely used in global navigation satellite systems (GNSS) such as GPS, where receivers calculate their position based on signals from multiple satellites, effectively performing trilaterasi in three dimensions.