Cone-angle parametrization of the array manifold in DF systems

The manifold of a planar array in a direction finding system may be considered as two families of azimuth θ and elevation ø curves, where the ø-parameter curves are hyperhelical as well as geodesic while the θ-parameter curves are neither. Since the θ-curves are not hyperhelical, their curvatures depend on θ and so analytical evaluation of curvatures of order greater than two can become exceedingly laborious and impractical. The advantages of having hyperhelical parameter curves are numerous. For one thing, all the curvatures of a hyperhelix may be evaluated recursively (since they do not vary from point to point) as a function of lower-order curvatures. This has been demonstrated in (1) for the case of the single-parameter manifold of a linear array. Furthermore the convenient nature of a hyperhelixʹs geometry has proven invaluable in array design (2), in investigating the detection and resolution thresholds (3) and in identifying ambiguities inherent in array configurations (4). In view of the above facts, it seems logical that an alternative parametrization of the manifold surface, which results in two sets of hyperhelical parameter curves, can provide a great deal of additional insight into the nature of planar array behaviour and design. In this investigation, such a parametrization is identified and its significance is demonstrated by a number of examples/applications. Furthermore properties, such as Gaussian and geodesic curvatures, are defined and their implications with regards to isometric mappings are discussed.