Far-field approximations to the Kirchoff-Helmholtz representations of scattered fields

The standard far-field approximation to the Kirchhoff formula for the field scattered by a flat metallic plate
S
of arbitrary shape is given by a certain surface (double) integral. This double integral can be reduced to a line integral evaluated around the boundary of S. Moreover, if
S
is a polygon, this line integral can be reduced to a closed form expression involving no integrations at all. The use of such line integral representations can easily reduce the costs of numerical calculation by orders of magnitude. If the integrands are to be sampled
p
times per wavelength to achieve an acceptable degree of precision, and if
A
is the area of
S
, then the numerical evaluation of the double integral requires
p^{2}A/\lambda^{2}
functional evaluations whereas the line integral only requires
p\sqrt{A/\lambda^{2}}
. If
S
is a polygon with
N
vertices, then only
2N
functional evaluations are required to evaluate the closed form expression with no quadrature error at all.