Nauch. apparat., 1987, 2, 2, 69–75 PL ISSN 0257-3881



Toruń Radio Astronomy Observatory, Nicolaus Copernicus University
ul. Chopina 12/18, 87-100 Toruń, Poland
* Institute of Astronomy, Nicolaus Copernicus University,
ul. Chopina 12/18, 87-100 Toruń, Poland

Exact analytical solutions for calculating the shape and area of shadows cast by feed support legs which are oriented arbitrarily with respect to the main axis of parabolic antennas are presented. An example is given for a 32 m radiotelescope planned for the Toruń Radio Astronomy Observatory. A FORTRAN subroutine for computing the shadow effective area is included.

One of the factors contributing to the gain reduction of parabolic antennas is the blockage of the incoming radiation by a subreflector or feed mount and its support struts. The amount of losses depends in the first place on actual structural design of a radio telescope, aperture illumination function and operating frequency. To the best of our knowledge there is no theory which would allow an exact account for these losses [1]. The usual practice is to find the shadow area, weighted by the aperture illumination, by graphical projection. This procedure is questionable when the dimensions of the blocking obstacles are of the order of a wavelength, because the radio shadow is then much wider than the optical shadow. For many reasons, however, the optical projection is very useful, first approximation to the real situation.

Most of the feed support structures in contemporary radio telescopes have the form of equiangular tri- or quadrupods whose vertices lie somewhere above the prime focus of a telescope. Ruze [2] has derived relatively simple formulae approximating the shadow area and gain loss for such strut configurations. More recently supports with struts skewed with respect to the main axis of the paraboloid have begun to appear. The 15 m millimeter UK/Netherlands telescope on Mauna Kea and one of the designs of a 32 m telescope planned for the Toruń Radio Astronomy Observatory are examples. In these cases the axis and strut no longer lie in the same plane. Quite naturally, literature is silent about practical procedures to calculate the shadow cast by such constructions. We set out to derive exact formulae for the shadow area, which cou1d be considered as both an improvement on, and generalization of Ruze's findings.

The optical shadow projected on the aperture plane consists of three parts [2]. Since the inner two parts can be dealt with straightforwardly by using conventional precedures, we shall limit ourselves to considering only the outer portion — the shadow of the struts caused by the spherical wave converging towards the feed or effective focal point. On the aperture plane the shadow is seen as composed of fan-shaped regions. Their area is determined by the orientation of the struts relative to the paraboloid axis, their lateral dimensions and the telescope parameters: radius rmax, and focal length f.

The analysis that follows refers to one fan-shaped region of the shadow (Fig. 1), corresponding to one of the legs of the support structure. Before going into details the reader will observe that the elongated shadow sides are determined by the section of two planes and the paraboloid of revolution. The planes pass through the prime focus of the paraboloid and are tangent to the strut on either side. Given the geometry and orientation of the strut it is a matter of elementary analysis to find directional cosines of both the planes. Let n denote versor normal to one of the plains so chosen that n3 is positive. Then on the aperture plane, (x, y) whose origin is at the focus, the equation of the shadow side corresponding to that plane will be

Fig. 1. Schematic drawing of aperture plane projection of a shadow cast by a skewed strut fixed to the parabolic dish surface at rmin. The shadow area is bounded by 4 circular arcs whose radii are rmin, rmax, R = 2f/n3 and R' = 2f/n'3

(x + 2fn1/n3)2 + (y + 2fn2/n3)2 = (2f/n3)2. (1)

This can be recognized as the equation of a circle. In the polar coordinates (r,φ) with the same origin the circle is conveniently described by either

φ(r) = φo – sign(φ o) arccos[(r2 – 4f2)/(2 r ro)] (2)

or its inverse
r(φ) = ro[cos(φ – φo) +


g2 – sin2(φ – φo)
ro = 2f


(n12 + n22)/n32,
   φo = arctg(n2/n1)
are coordinates of the center of the circle (1) and g2 = (2f/ro)2 + 1. The quadrant for φo is determined from relative signs of –n2 and –n1. This simple analysis can be repeated for the second tangent plane. So, for the sake of introducing the notation, starting with n′ leads to φ′(r) and r′(φ) with parameters r'o, φ′o and g′. The effective area of the shadow can be defined as

S = rmax

[φ(r) – φ′(r)] f(r) r dr

where f(r) stands for the illumination function and rmin is the radial distance of a point, at which the strut intersects the dish surface. Though, unfortunately, the function (2) is not directly integrable (since it leads to the elliptical integral) the integral (4) can be usefully evaluated numerically, given the f(r). Nevertheless, S can be obtained analytically using function (3) and for some simple forms of f(r). For brevity we shall omit here the details of somewhat tedious derivation and present only the results of such an approach. Let

U(α,g) = {g P(α,g) + sin[P(α,g)]} 4f2/(g – 1/g), (5)
P(α,g) = α + arcsin[sin(α)/g] (6)

then the projected area of shadow region (with f(r) = 1) is given by

So = 0.5 [r2max2 –φ 2′) – r2min1 – φ1′) – U(φ2 – φo,g) +
+ U(φ1 – φo′,g) + U(φ2′ – φo′,g′) – U(φ1′ – φo′,g′)], (7)

where φ2 = φ(rmax), φ1 = φ(rmin) and the primed quantities refer to using φ′(r) instead of φ(r). If an illumination taper is assumed to be of the form f(r) = 1 – a(r/rmax)2, the effective area can be found from

S = So – (0.25 a/r2max)[r4max2 – φ2′) – r4min1 – φ1′) –
– D(φ2 – φo,g) + D(φ1 – φo,g) + D(φ2′ – φo′,g′) – D(φ1′ – φo′,g′)], (8)
D(α,g) = {(g2 + 2)P(α,g) + [3g + 1/g + cos(P(α,g))]×
×sin(P(α,g))}(2f)4/(g – 1/g)2.

Formulae (5) through (9) represent the exact analytical solution to the problem of finding the shadow area of skewed feed support legs, as projected on the aperture plane. Although there should be no difficulty to program the presented algorithm, for the convenience of the reader, we enclose an Appendix with a listing of a FORTRAN subroutine SHADOW along with an example of a simple calling program. We have checked the subroutine against numerical integration carried out according to (4). Here follows an example from our study of the mentioned Toruń design.

The 32 m telescope (rmax = 16 m) is assumed to have the focal length f = 11.2 m and feed support composed of 8 cylindrical legs, each 0.159 m in diameter, and each attached so that it casts a shadow of the same shape and area. Two of (x, y, z) points, through which the axis of one of the struts passes are (5.719, 0.0, –10.5764) and (2.1213, 2.1213, 0.38) (in meters).

Fig. 2. The outer part of the feed support blockage area of the Toruń 32 m radio telescope. The figure is to scale. The dashed region is here 11.28 m2

This gives 5.6869 m for rmin. The resulting shadow shape is shown in Fig. 2 and its area calculated in accordance with (7) is 5.64 m2, which is 6.42% of one eighth of the aperture area outside rmin. With a typical 12dB illumination taper (a = 0.75) the effective area falls to 3.2 m2, which translates to 6.31% of the effective area outside rmin (8×50.73 m2).


K. M. Borkowski would like to gratefully acknowledge a financial support from Polish Academy of Sciences through research problem Nr. l02a.


1. RUSCH W. V. T., Analysis of Paraboloidal — Reflector Systems, Methods of Experimental Physics, 12B, 29 (1976).
2. RUZE J., Feed Support Blockage Loss in Parabolic Antennas, Microwave J.,11, 12, 76 (1968).



c     Sample calling program for subroutine SHADOW
      dimension xyz1(3), xyz2(3)
      data xyz1,xyz2/5.719,0.,-10.5764,2.1213,2.1213,.38/
      data f,rleg,taper,rmax/11.2,.0795,.75,16./
      call SHADOW(xyz1,xyz2,rmax,f,rleg,taper,areaeff)                  
      write(6,11111) areaeff
11111 format(//1x,'Effective area = ', f10.6)     

      subroutine SHADOW(xyz1,xyz2,rmax,f,rleg,taper,areaef)
c     This subroutine calculates an effective area of the shadow
c     cast by a feed support leg arbitrarily oriented with respect
c     to the main axis of a parabolic reflector.
c     Description of parameters:
c         xyz1, xyz2 - one dimensional arrays containing the three
c                      cartesian coordinates of two points lying on
c                      the leg axis.
c               rmax - the reflector radius.
c                  f - the focal length of the paraboloid.
c               rleg - the radius of the leg
c              taper - the illumination taper parameter. Illumination
c                      function is assumed to have the form:
c                             1 + taper*(r/rmax)**2    
c                      where r is the distance from the paraboloid axis.
c             areaef - the effective area of the shadow as projected on
c                      the aperture plain.
c     Remarks
c     The units for xyz1, xyz2, rmax, f and rleg are arbitrary but the
c     same. Areaef is output in these units squared.
c     It is assumed that  xyz1(3) .lt. xyz2(3) .

      double precision xn(3),xn1(3),xn2(3),r1(3),r2(3),a,b,c,D,del,
      dimension xyz1(3),xyz2(3)
c     Statement functions
     1         (ff*ff/(g-1/g))**2
c     Find the lower limit of the shadow
c     Rotate the focal plane.
      r1(1)= c*xyz1(1)+s*xyz1(2)
      r2(1)= c*xyz2(1)+s*xyz2(2)
c     Determine vector normal to the axis of leg
c     Determine vector normal to plains tangent to the leg
      do 10 i=1,3
   10 continue
c     Find polar parameters of the shadow
      f0 =datan2(-xn1(2)*xn1(3),-xn1(1)*xn1(3))
      r0 =ff*dsqrt(xn1(1)**2+xn1(2)**2)/dabs(xn1(3))
      g =dsqrt((ff/r0)**2+1.d0)
c     Find the shadow area and its effective area
     *          +U(f2p-f0p,gp)-U(f1p-f0p,gp))
     *          -D(f2-f0,g)+D(f1-f0,g)+D(f2p-f0p,gp)-D(f1p-f0p,gp))))

Note on this electronic version of the paper

A few mistakes or misprints in the published version were corrected. These were generally minor ones but there was a more significant error that made the Eq. (8) therein invalid (omitted term a/r2max); luckily the subroutine SHADOW was not affected. This subroutine was now (in 2002) reconstructed (the original has been lost) for the purpose of including in the present web publication and compiled with the Lahey compiler LF90 (v. 4.5) with no problems whatsoever. Executing the compiled code should result in the following response:
Effective area = 3.202006
(possible difference in the least significant digit should not be considered alarming). However, calculation of rmin in this routine is only approximate, and although adequate for this example it should not be recommended in general.

More recently (in 2012) the SHADOW routine has been tested extensively and in the process improved considerably (exact calculation of rmin inclusive). Its new version, SHADOW2 (see below), may be of interest to wider readership not only because of the mentioned improvement, but also because it returns, besides areas, all parameters necessary for analytical description of shadows.

The example in the following printout returns such data (see the commented out end part of the main program) for Torun 32-metre telescope, which has each strut skewed and composed of two cylindrical parts of different diameter. The reader may just mark this FORTRAN listing, copy it to the clipboard and save to a file to have a source code ready for modifications and compilation. Since it is written in a pretty standard language it should be smoothly digested by a variety of compilers. Here (see figure Rys. 4 therein) is a report that used results from this example to compute quite realistic (although also quite complex) radiation patterns of the telescope.

Be aware that the algorithm employed in SHADOW and SHADOW2 cannot tackle the trivial case of support legs parallel to the paraboloid axis (because then the shadow circular sides degenerate to straight lines, or circles of inifinite radius). However, this case can be easily dealt with directly as the shadows become just radial sections of circular aperture.

      program Example_Usage

      implicit real*8 (a-h,o-z)
      dimension xyz1(3), xyz2(3), xyz3(3)

c Data for Torun 32-m radio telescope
      data xyz1,xyz2/5.719,0.,-10.5764,2.1213,2.1213,.38/,
     &     f,rleg,rlegc,taper,rdish/11.2,.0795,.057,.75,16./

c Coordinates of beginning of thinner part of support strut
      do 1 i=1,3
1     xyz3(i) = xyz1(i) + (xyz2(i) - xyz1(i))*2.28/(1.97 + 2.28)

c Calculate parameters for thicker part (inner, up to xyz3 point)
      rmin = 0
      rmax = -rdish              ! negative to calculate area below xyz3
      call SHADOW2(xyz1,xyz3,rmin,rmax,f,rleg,taper,
     & r0,phi0,r0p,phi0p,ar,areaeff) 
      R  = dsqrt(r0**2  + 4*f*f)
      Rp = dsqrt(r0p**2 + 4*f*f)
      write(6,"(/1x,4(a,f8.4)/a,1p,2e15.6)") 'Area:',ar,
     &', Effective area:',areaeff,', rmin:',rmin,', rmax:',rmax,
     &"  Shadow side radii, R and R':",R,Rp
      print'(a,2(f15.6,f9.6))',"  r0,phi0, r0',phi0':",r0,phi0,r0p,phi0p

c The same for thinner part (outer, from xyz3 point to end of shadow)
      rmin = -1                  ! negative to calculate area above xyz3
      rmax = rdish
      call SHADOW2(xyz3,xyz2,rmin,rmax,f,rlegc,taper,
     & r0,phi0,r0p,phi0p,ar,areaeff) 
      R  = dsqrt(r0**2  + 4*f*f)
      Rp = dsqrt(r0p**2 + 4*f*f)
      write(6,"(/1x,4(a,f8.4)/a,1p,2e15.6)") 'Area:',ar,
     &', Effective area:',areaeff,', rmin:',rmin,', rmax:',rmax,
     &"  Shadow side radii, R and R':",R,Rp
      print'(a,2(f15.6,f9.6))',"  r0,phi0, r0',phi0':",r0,phi0,r0p,phi0p

c ======== Results of this Example as displayed on the screen ========

c Area:  0.7717, Effective area:  0.3385, rmin:  5.6868, rmax:  8.1744
c  Shadow side radii, R and R':   6.497087E+01   6.596239E+01
c  r0,phi0, r0',phi0':      60.987322 2.337901      62.042540 2.282220

c Area:  3.4902, Effective area:  1.8229, rmin:  8.1744, rmax: 16.0000
c  Shadow side radii, R and R':   6.509847E+01   6.580912E+01
c  r0,phi0, r0',phi0':      61.123245 2.330007      61.879559 2.290087


      subroutine SHADOW2(xyz1,xyz2,sRmin,sRdish,f,rleg,taper,
     & r0,phi0,r0p,phi0p,area,areaef)

c Calculates shape and effective area of the shadow cast by a feed 
c support leg arbitrarily oriented with respect to the main axis of 
c a parabolic reflector of a radio telescope. Here tha basic algorithm
c is taken from KM Borkowski, AJ Maciejewski, Scientific Instrumentation,
c vol. 2, p. 69-75 (1987). 
c See:

c  Units of lengths quantities are arbitrary (but the same);
c  output areas are in these units squared and angles are in radians.
c xyz1, xyz2 - arrays containing the three Cartesian coordinates of
c              two points on the axis of leg; xyz1 is closer to dish
c              or such that z1 = xyz1(3) < z2 = xyz2(3), z having
c              origin in prime focus and being positive away from dish;
c              avoid an axis parallel to z-coordinate (as shadow circle
c              radius becomes then infinite) - make it slightly slant
c sRmin      - signed radius of beginning of shadow area to calculate;
c              if sRmin < 0, this is substituted by radius of shadow of
c              xyz1 (xyz1 projected on the dish from prime focus); if
c              0 < sRmin < rCross, rCross is taken  for the beginning
c              (rCross is radius where the axis crosses the paraboloid);
c              on output sRmin equals rmin assumed for area calculation;
c              these various assigments, along with those of sRdish, ease
c              analyses of support legs composed of different segments;
c              one may begin with sRmin = 0 and dish radius in sRdish;
c              on output SRmin contains rmin used for area calculation
c sRdish     - signed main reflector radius; if sRdish < 0, the upper
c              limit for shadow area calculation is taken as radius
c              of shadow of xyz2 on the dish (as projected from focus);
c              on output SRdish contains rmax used for area calculation
c f          - focal length of the paraboloid
c rleg       - radius or half-width of the leg as seen from prime focus;
c              the leg is assumed cylindrical, so if it is rectangular
c              of width w, take rleg = w/sqrt[4 + (w/d)**2], where d is
c              the distance of the axis of leg wall facing the prime
c              focus (as given by xyz1 and xyz2) from the prime focus
c taper      - illumination taper parameter; assumed illumination
c              functions is:   1 + taper*(r/sRdish)**2,    
c              where r is the distance from paraboloid axis
c r0, phi0   - polar coordinates, radius and angle, of centre of circle
c              whose section constitutes one of the shadow sides; 
c              the circle equation is:
c    phi(r) = phi0 - sign(phi0) acos[r/(2 r0) - 2 f**2/(r r0)], or
c    r(phi) = r0 cos(phi - phi0) + sqrt[2f**2 + (r0 cos(phi - phi0))**2]
c r0p, phi0p - as above but for the other side of shadow     
c area       - area of the shadow as projected on the aperture plain
c areaeff    - same but weighted with the illumination function (taper)

      implicit real*8 (a-h,o-z)
      dimension xyz1(3),xyz2(3),xn(3),xn1(3),xn2(3)

c Definitions of functions used in area calculation only
      p(a,g)  = a + dasin(dsin(a)/g)
      u(fi,g) = (g*p(fi,g) + dsin(p(fi,g)))*4*f*f/(g - 1/g)
      d(fi,g) = ((g*g + 2)*p(fi,g) + (3*g + 1/g + dcos(p(fi,g)))*
     &          dsin(p(fi,g)))*(4*f*f/(g - 1/g))**2

      rmax = abs(sRdish) ! this is initial upper limit for shadow;
c lower limit, rmin, will be determined by sRmin or leg axis (straight
c line through xyz1 and xyz2) cross point on the dish as follows
c Equation of parabola: r=2f*tan(theta/2) or r**2 = x**2+y**2 = 4f(f+z),
c Equations of the axis:  z-z1 = (x-x1)*dz/dx, x-x1 = (y-y1)*dx/dy
c Solution (zCross) to these three equations follows
      x1 = xyz1(1)
      y1 = xyz1(2)
      z1 = xyz1(3)
      dx = xyz2(1) - x1
      dy = xyz2(2) - y1
      dz = xyz2(3) - z1
	drSqr = dx*dx + dy*dy
      z0 = 2*f*dz**2 - dz*(dx*x1 + dy*y1) + drSqr*z1
      zo = Sqrt(dz**2*(4*dz**2*f**2 - 4*dy*dz*f*y1 + 
     &   dx*(2*dy*x1*y1 - 4*dz*f*x1) + dy**2*(4*f*(f + z1) - x1**2) 
     &   + dx**2*(4*f*(f + z1) - y1**2)))
      zCross = (z0 - zo)/drSqr
c     zCross2= (z0 + zo)/drSqr            ! this is the other solution
      rCross = sqrt(4*f*(zCross + f))
      rmin   = min(max(sRmin,rCross),rmax)

            if( then ! take rmin = r(xyz1_shadow)
      theta = datan2(sqrt(xyz1(1)**2 + xyz1(2)**2),-xyz1(3))
      rxyz  = 2*f*dtan(theta/2)
      rmin  = max(rxyz,rmin)
            if( then ! take rmax = r(xyz2_shadow)
      theta = datan2(sqrt(xyz2(1)**2 + xyz2(2)**2),-xyz2(3))
      rxyz  = 2*f*dtan(theta/2)
      rmax  = min(rxyz,rmax)

c Vector normal to the plain passing through leg axis and prime focus
      xn(1) = xyz1(2)*xyz2(3) - xyz2(2)*xyz1(3)
      xn(2) = xyz1(3)*xyz2(1) - xyz2(3)*xyz1(1)
      xn(3) = xyz1(1)*xyz2(2) - xyz2(1)*xyz1(2)

c Vectors normal to similar planes but tangent to the leg sides
      dx1x2= xyz1(1)*xyz2(1) + xyz1(2)*xyz2(2) + xyz1(3)*xyz2(3)
      dx1  = xyz1(1)*xyz1(1) + xyz1(2)*xyz1(2) + xyz1(3)*xyz1(3)
      dx2  = xyz2(1)*xyz2(1) + xyz2(2)*xyz2(2) + xyz2(3)*xyz2(3)
      tak=rleg/dsqrt(dx1*dx2 - dx1x2**2 - rleg**2*(dx1 - 2*dx1x2 + dx2))
        do 1 i = 1,3
      del    = ((dx2 - dx1x2)*xyz1(i) + (dx1 - dx1x2)*xyz2(i))*tak
      xn1(i) = xn(i) + del
1     xn2(i) = xn(i) - del

c Polar parameters of the shadow sides (angles and distances [from
c dish axis] of centres of circles corresponding to the sides)
      phi0  = datan2(-xn1(2)*xn1(3),-xn1(1)*xn1(3))
      phi0p = datan2(-xn2(2)*xn2(3),-xn2(1)*xn2(3))
      r0    = 2*f*dsqrt(xn1(1)**2 + xn1(2)**2)/dabs(xn1(3))
      r0p   = 2*f*dsqrt(xn2(1)**2 + xn2(2)**2)/dabs(xn2(3))

c Make sure that the two triangles to solve are really triangles,
c i.e. that rmin + r0 > R = 2*f/n3 = sqrt(r0**2 + 4*f**2) and
c rmax < R + r0 = sqrt(r0**2 + 4*f**2) + r0
      rmi = max(dsqrt(r0**2 + 4*f**2) - r0,dsqrt(r0p**2 + 4*f**2) - r0p)
      rmx = min(dsqrt(r0**2 + 4*f**2) + r0,dsqrt(r0p**2 + 4*f**2) + r0p)
      rmin = max(rmin,rmi*1.00000001d0) ! 1d-8 is to avoid machine error
      rmax = min(rmax,rmx*0.99999999d0) !      when r = R +/- r0 exactly
      sRmin  = rmin                     ! return the value actually used
      sRdish = rmax                     ! return the value actually used

c Shadow sides can now be described by these equations
c phi(r)  = phi0  - sign(phi0) acos[r/(2 r0)  - 2 f**2/(r r0)] and
c phip(r) = phi0p - sign(phi0p) acos[r/(2 r0p) - 2 f**2/(r r0p)]
c for rmin <= r <= rmax

c Prepare for computatation of the shadow area
      tmp1 = (rmin - 2*f)*(rmin + 2*f)/(2*rmin)
      tmp2 = (rmax - 2*f)*(rmax + 2*f)/(2*rmax)
c f1, f1p, f2p and f2 are angular coordinates of the four shadow corners
      f1   = phi0  - dsign(dacos(tmp1/r0),phi0)
      f2   = phi0  - dsign(dacos(tmp2/r0),phi0)
      f1p  = phi0p - dsign(dacos(tmp1/r0p),phi0p)
      f2p  = phi0p - dsign(dacos(tmp2/r0p),phi0p)
      df1  = f1-f1p              ! angular width of shadow at rmin
      df2  = f2-f2p              ! angular width of shadow at rmax
      g    = dsqrt((2*f/r0 )**2 + 1)
      gp   = dsqrt((2*f/r0p)**2 + 1)

c Geometrical and effective area of shadow
      area = .5*(rmax**2*df2-rmin**2*df1-u(f2-phi0,g)+u(f1-phi0,g)
     &     +u(f2p-phi0p,gp)-u(f1p-phi0p,gp))
      areaef = dabs(area-(.25*taper/sRdish**2)*(rmax**4*df2-rmin**4*df1
     &     -d(f2-phi0,g)+d(f1-phi0,g)+d(f2p-phi0p,gp)-d(f1p-phi0p,gp)))

File originally translated from TEX by TTH, version 3.13 on 25 Sep 2002. Last modified: 19 July 2012