Afstandsforskelle

Afstandsforskelle is a term that describes the differences between distances measured from one or more reference points. In mathematics and applied sciences, it is often used to express how much one distance differs from another, for example the difference between the distances from a moving point to two fixed points.

Geometrically, the difference of distances to two fixed points defines a conic section. If A and B are fixed and P is a variable point, the set of points P for which the absolute difference |PA − PB| is constant equals a hyperbola with foci at A and B. If the constant is zero, the locus becomes the perpendicular bisector of AB. These properties underlie many distance-based constructions in geometry and navigation.

Applications of afstandsforskelle include navigation, surveying, triangulation and trilateration, as well as pattern recognition and geolocation techniques. In GPS and differential positioning, differences in measured distances to multiple reference points are used to determine location while mitigating common-mode errors such as atmospheric delays.

In metric spaces, the concept connects to the reverse triangle inequality: for any points x, y and a, the inequality |d(x,a) − d(y,a)| ≤ d(x,y) holds. This provides bounds on how much a distance to a reference point can change when the source point moves.

Computation typically relies on Euclidean distance using the Pythagorean theorem, though the idea extends to general metrics. Practical use must address measurement noise, potential ambiguity from intersecting distance curves, and the need for additional measurements to obtain a unique solution.

See also: distance, metric, hyperbola, triangulation, trilateration, differential GPS.