Finding Minimum Values of Distances and Distance Ratios in ATS Spectrum Management

Preventing interference among radio circuits used for air traffic services (ATS) often requires spectrum managers to observe channel-assignment rules based on minimum ratios of the great-circle distances (GCDs) traversed by desired and undesired (potentially interfering) signals. Each circuit operates within a service volume of airspace having a circular or polygonal "footprint" on the surface of a spherical Earth. Minimizing undesired-to-desired GCD ratios with respect to these footprints can be significantly facilitated in many cases through the use of stereographic projection. Instead of performing the minimization using the footprints on the surface of the sphere, a stereographic projection of the footprints to the complex plane is performed to transform the original minimization problem into a simpler problem of minimizing a ratio of distances in the complex plane. This ratio can be expressed in terms of a single real variable and then minimized using the Newton-Raphson method.