The formula for z which you gave would give me nightmares.
Are you absolutely sure that what you have written is correct?
Perhaps first principles was overstated n my posting

Art

"Richard Clark" wrote in message
...
On Fri, 11 Mar 2005 23:05:28 GMT, "
wrote:

A cardioid pattern has radiation in the 180 degree portion behind the feed
point
first described by Johann Castillon made a Fellow of the Royal Society
of London in 1753:
r = 2a(1 + cos(theta))
where theta = 180 such that:
r = 2a(1 + cos(180))
r = 2a(1 + -1)
r = 0

Hi Art,

Understandably, the term you are so unfamiliar with, insofar as no
rearward radiation (not the same as no radiation to 180 degrees) is
Lambertian. As may be expected, the term is derived from the work of
mathematician Johann Heinrich Lambert (1728-1777).

It is a distribution curve derived from reflections off of a "diffuse
surface" (note, not the same thing as reflections off of a specular
antenna element):
(x²+y²+z²)² = z

At 300 years+ both are pretty old works that each easily qualify
within the purview of "first principles" if one is serious about
radiation.

73's
Richard Clark, KB7QHC