I also tried different heights off of the reflector. I varied the distance from 1/16 to 1/4, with the impedance being better at 1/8th. I had eliptical shapes and some very odd ones, but they all worked as well. The shape of the loop, and where it gets feed, does change the impedance. Practically, these antennas worked equally as well, though given the readings I have now made, some of these shapes have a horrible impedances.
We hadn't originally taken into account the velocity factor of the wire in the antenna. i.e. the speed of light in the wire is less than in air. Having a slower speed in the wire, means that the frequency shifts downward (frequency = velocity/wavelength), so the wire needs to be shorter to resonate at the same frequency. Measuring a 2.442GHz antenna, 20 year on, shows it was tuned better to 2.35GHz, so was too long.
The velocity factor is going to vary with different wire thicknesses. In our case the overall wire length should be reduced by 4% to 5%. Plastic close to the wire will change the velocity factor too, further reducing the speed of the radio wave. This happened to us, when we put the antenna inside a plastic case. Copper oxide on the wire will have this effect too, so an aging antenna is going to slowly detune to a lower frequency.
The velocity factor represents the percentage slowdown of the incoming signal in air, compared to the signal in the copper wire.
This is the same affect you see with light going through water or glass (light being a radio wave).
The light induces a wave in the electrons in the glass, but the speed of that induced wave is less than the speed of light in air.
If you add both the original light's radio wave, and the slower induced wave in the material, you get a resultant wave that we see.
When the light leaves the glass it speeds up again. The light isn't magically gaining energy.
It has just stopped interacting with the electrons in the material.
The material doesn't need to be conductive either, just a dialetric like plastic.
In a dialectric, the radio wave alternates the polarization (alignment of charge) in the material, but no current flows.
See also BiQuad Calculator
These have long been retired, but I got a nanoVNA recently, and decided to go back and see how well tuned these antenna were. They all had been working 500m to 2.5km, so from a practical point of view, they were great antenna.