Download search solid angle equation (4)

APPENDIX I – DERIVATION OF SEARCH SOLID ANGLE EQUATION
R cos 
dr, d
R

ds, d 

Figure 1 – Geometry for Computing Solid Angle
We can write the area of the small square in the figure above as
dA  dsdr   R cos  d   Rd 
or
dA  R 2 cos  d d  .
If we want the total area over the angles 1,2   0   2,0   2 and
0  
A
2, 0   2 we can integrate the above over these angle ranges, viz.
0  2 0  2

0 

2 0 
R 2 cos  d  d .
2
Performing the integral results in
A  R 2 sin 0   2   sin  0   2   .
After some trigonometric manipulations this reduces to
A  2R2 cos 0 sin   2  .
Finally, dividing by R 2 yields the solid angle, viz.
  2 cos0 sin   2  .
©2005 M. C. Budge, Jr