Aircraft proximity termination conditions for 3D turn centric modes

International airspace design is undergoing significant change that requires formal and rigorous mathematical specifications to assure the safety of flight operations. Aircraft proximity management is one such area. The Point of Closest Approach (PCA) between aircraft flightpaths is the position along a flightpath at which the minima in relative range occurs. To date, PCA has been estimated primarily based on the assumption of a linear extrapolation of the velocity vectors of each aircraft involved, however, this assumption is limiting and aircraft in turning flight must also be considered. A generalised geometric and vector construction for the determination of PCA is presented. A solution based on a characterization of Fermat’s method for stationary points is presented that results in a complex transcendental equation. By casting the equation in a determinant structure a co-linearity condition is revealed between three unique 2D points. A novel aspect is that one of these points is a fixed reference point that lies on either the vector between the aircraft turn centres or on one of its extensions providing a reference to determine the location of the PCA. The analytic method can be readily applied in a laboratory test environment or in an automated guidance context. The rigorous proof enables a higher confidence in achieving dependable operations in a safety critical context and in the adequacy of test strategies when developing algorithms for aircraft avionics.