Conflict Management: Apollonius in airspace design

•Analytic extensions to the Apollonius geometric theorem.•Generalised Barycentre concept for pair-wise proximity.•New conflict detection paradigm – Non-Allowed Steering Direction (NASD).•New path-isometry for Earliest Intercept Point (EIP) guidance law.•3D surface for force field control law singularities.

Conflict detection is the process by which potential collisions between aircraft are identified. Developing efficient detection methods is an engineering priority since the prevention of a mid-air collision is a major safety constraint for all airspace design. Proximity between two aircraft each flying with instantaneous uniform straight motion is re-examined. The Apollonius theorem for constructing a circle shows that for a constant speed ratio the locus of all collision points is a circle in the 2D conflict plane (generalised to a sphere in 3D space). This circle provides an effective method and reference for determining the set of all 3D non-allowed steering directions (NASD), which, if followed by ownaircraft, would result in either a collision or proximity of operational concern. The NASD paradigm permits the identification of a protection zone encapsulating the region of potential collision. The method has been applied in the design of static airspace structures and accident analysis as well as providing an explicit solution for the determination of a surface of singularities that are characteristic of force-field guidance laws. The NASD paradigm provides deeper insight into the conflict detection process and represents an increase in the utility, diversity and assessment of design options for future systems.