By Athanassios Manikas

In view of the importance of the array manifold in array processing and array communications, the function of differential geometry as an analytical software can't be overemphasized. Differential geometry is principally limited to the research of the geometric houses of manifolds in 3-dimensional Euclidean area R3 and in actual areas of upper measurement.

Extending the theoretical framework to complicated areas, this beneficial booklet offers a precis of these result of differential geometry that are of functional curiosity within the research of linear, planar and 3-dimensional array geometries.

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**Sample text**

However, parametrization in terms of the arc length s (see Fig. 2), which is the most basic feature of a curve and a natural parameter representing the actual physical length of a segment of the manifold curve in July 6, 2004 9:29 WSPC/Book Trim Size for 9in x 6in 24 chap02 Diﬀerential Geometry in Array Processing CN , is more suitable. There is a further advantage of using s as a parameter: the arc length s (in contrast to p) is an “invariant” parameter. This means that the resulting tangent vector to the curve, expressed in terms of s, always has unity length.

9), forms the matrix U(s) = [u1 (s), u2 (s), . . e. e. 10) The matrix F(s) is a continuous diﬀerential real transformation matrix called frame matrix. 1. Fig. 9 “Moving frame” U(0) and U(s). July 6, 2004 9:29 WSPC/Book Trim Size for 9in x 6in chap02 Diﬀerential Geometry of Array Manifold Curves 29 Properties of the frame matrix F(s). 3 Frame Matrix and Curvatures The question is, how the frame matrix F(s), at the running point s, is related to the curvatures of the manifold attached to this point.

The constant ﬁrst curvature of the corresponding isotropic array is also shown in the same ﬁgure. It is seen that there exist two bearings where the ﬁrst curvature becomes equal to zero. From the variation of the ﬁrst curvature, it is possible to deduce the geometrical object to which the manifold best ﬁts. In the case considered, it is apparent that the eﬀect of the sinusoidal elemental pattern is to deform the hyperhelix to a geometrical ﬁgure resembling an “eight” with a double point at the origin of the coordinates of CN .