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As seen in Fig. 7, displacement of the hyperboloid normal to and at an angle with the design focal axis causes an effective displacement of the RF source image from the focus of the design parabola. The beam deviation angle, θhr, is slightly increased by M, the magnification factor, and is dependent on α, the angle between the design focal axis and the rotated hyperbola axis.

The beam-deviation or pointing-error equation is dependent on the location of the virtual focal point, A. This can be found as follows:

First, displace the feed an amount equal to the displacement of the hyperbola vertex (-δhr).

The initial displaced focal point, C, = N tan α, and the beam deviation angle is:

For small angles,

Therefore,

A correction must be applied to relate the displaced feed point to the design focal axis. This is accomplished by “translating” the displaced focal point an amount equal to +δhr back to the design focal axis. This latter effect is also identical to beam deviation due to feed translation. Therefore:

Since,

the resultant beam-deviation or pointing-error equation for hyperbola rotation becomes:

From the virtual focal point, A, the same equation can be developed as follows:

A = B + C.

Since,

then,

Since,

The effect of hyperbola rotation on the pointing angle is shown in Fig. 8. Note that the beam deviation angle, θhr, is approximately one-tenth of the actual rotated hyperbola angle, α.