Quite a few years ago, there were a couple of articles -- in Ham Radio
magazine, I believe --, which addressed that topic. As I recall, the
author used an optimization program that allowed the shape to vary. He
ended up with elements bent kind of like a gull wing, and considerably
longer than a half wavelength -- more like a wavelength if I recall
correctly. The gain was substantially more than for a Yagi, but I don't
remember how good the F/B ratio was. I believe he did construct and
measure some actual antennas.

I'm sorry I don't recall the name of either the author or the articles.
Hopefully, though, this posting will spur someone with a better memory
(and that includes just about everyone) to recall and report enough
information for you to find the articles.

The articles were very interesting, but it looked to me like the
construction problems would outweigh the benefits. Apparently I'm not
the only one to reach that conclusion, because in the intervening years
I've never seen an actual antenna resembling that design.

Roy Lewallen, W7EL

Dan Jacobson wrote:

Just curious, what is the result of computer modeling for the ideal
shape of Yagi elements? I couldn't just be simple tubes.