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{zCos[e] + x Cos [f ] Sin [ e ] + y Sin [e] Sin[f] xCos[e] Cos [f ]  z Sin [e] + y Cos [e] Sin[f] y Cos[f]  x Sin[f]}
Cos[f] Sin[e] Cos[e] Cos[f]  Sin[f] Sin[e] Sin[f] Cos[e] Sin[f] Cos[f] Cos[e]  Sin[e] 0
The results (output) are the product of Eq. (2.1) and the transposed matrix of matrix A, respectively.
The equation of the paraboloid is a) in spherical coordinates p = 2 f / (1 + cos y) = f sec2 (y / 2) (2.4)
b) in cartesian coordinates r2 = x2 + y2 = 4 f(f + z)
where the paraboloidal coordinates (u, v, x) are connected to the cartesian ones by and c is the azimuthal angle about the zaxis.
In this coordinate system the paraboloid's surface is described by u = constant (or v = constant). At the vertex V we have v = 0, hence z = y u2 = f and thus u = V"2 f. Also dp = vdv for u = constant. The infinitesimal surface element is dS = VU2 + v2 uvdv d^ = 2Vf p dp dc = 2cos(2)p dp dc (2.7)
where we have used Eq. (2.4). On eliminating p we obtain for the surface element and the surface of the paraboloid with aperture halfangle Y0 follows as the integral of Eq. (2.8)
f2 A sec3 ( YM  1] = 8t f2 [{1 + (df M2}T 1]. (2.9)
Here we have used the alternative expression tan(y/ 2) = r / 2 f, (r2 = x2 + y2),
which is easy to derive from Eq. (2.4), considering that sin y = r / p. From this also follows tan ( 2 M = 4f. (2.10b)
The angle Y0is called the "aperture (half)angle" of the paraboloid, i.e. the angle between the axis and the "edgeray" from FP. The "depth" D of the paraboloid is given by w = f ( ff M2 = f (tan ^ )2 (211)
The following relations also hold r/ f sin y=5, (2.12)
The formulas for the secondary reflectors in spherical coordinates are hyperboloid (Cassegrain) rs = (c2  a2 )/(a + c cos y) (2.14)
ellipsoid (Gregorian) rs = (c2  a2 )/(c  a cos y) (2.15)
where c and a are the usual parameters describing the conic sections (see above).
The Cassegrain system is characterised by the magnification factor m, connected to the eccentricity e = c / a of the hyperboloidal secondary by the relations m = (e + 1) / (e  1) (2.16a)
The "equivalent paraboloid" of the Cassegrain antenna is given by tan f__r_
tan 2 = 2 m f from which follows tan F = Tf , (2.17)
2 4mf' v y where F0 is the opening halfangle of the secondary reflector seen from the Cassegrain focus. Hence we have m = tan ( Y )/tan ( % ) (2.18)
where c is the "focal length" and ds the diameter of the hyperboloid. The distance between primary and secondary focus of the Cassegrain system is fc = 2 c and the distance from primary focus to secondary vertex is l = c  a (see Fig. 2.1). The following relations hold fc = 2c l = c  a = c (e  1)/e a = c / e = c (m  1)/(m + 1)
In the special case where the secondary focus coincides with the vertex of the primary reflector (i.e. f = f), we have the situation that Eq. (2.17) can be written as tan HFo / 2) = ds / 4 f, leading to the simple expression m = dp / ds.
Parameter 
Symbol 
Magnitude 
Prime reflector diam. 
dp 
12 m 
Primary focal length 
fp 
4.8 m 
Primary focal ratio 
fp / dp 

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