Art wrote:
"The computer program is built on those mathematics, and an antenna
program will ALLWAYS produce radiators in equalibrium which means at an
angle."

Arnold B. Bailey disagrees in "TV and Other Receiving Antennas". On page
367 he writes:
"The directional action of a rod antenna best can be analyzed by
considering the rod as consisting of many tiny sections, connected
together to form a metallic circuit. A typical small segment X - X is
shown in Fig. 7-28 B; its position in a half-wave center-fed antenna is
indicated in part (A) of the figure. Each tiny section may be taken
sufficiently short compared to a wavelength so that the electromagnetic
wave acts practically instantaneously throughout one section, and hence
induces a substantially uniform current in that section. Such a short
antenna segment has a simple directional response pattern, indicated in
Fig. 7-28B, which is basic for all directivity calculations, since all
antenns may be considered to be made up of these tiny segments. This
fundamental response pattern varies as the cosine of the angle (which we
shall call theta) between the direction of the incoming wave and the
perpendicular through the center of the segment X - X, as indicated in
part (B) of the figure. If E stands for the value of the field intensity
(strength of the electric vector), then we can characterize the
directional response by the relation Ecos theta, which gives us the
relative magnitude of E for any wave direction relative to the antenna."

You probably have seen the figure-eight pattern of a dipole antenna and
are already aware that maximum response is broadside to the antenna at
its center. If the antenna is tilted away from the perpendicular its
response is diminished. Other antennas have a similar response as all
are made up of elemental segments.

Best regards, Richard Harrison, KB5WZI