Yagi is/was some Japanese professor who calculated the required relation of the elements.

First we need some wave theory, waves are symmetrical, a radio transmitter generates a wave shaped electrical potential on the output terminal. Guiding this potential over a wire will give the wire a potential over the length. As the wave travels with (near) lightspeed it is repeated every 'wavelength' on the wire. If we measure on the wire at wavelength distance we will measure nothing, the wave has identical potential between these points.

If we take the system of the transmitter and the antenna cable we can make the cable sing. That is we can make the length resonating with the wave by making it a multiple of the wavelength. The open end of the cable will be a crest in the wave as well as the last amplifier of the transmitter. By now making a tap at a half wavelength before the end we have the other polarity potential crest as well.

We now have a symmetrical (mirrored) wave on the tap and end. If we connect these to two identical half wave antennae and place these pointing away from each other we have gained some 3dB in signal. This is doubling the signal strength at no energy cost . This consrtuction is a so called open dipole antenna.

The electromagnetic waves can not penetrate a Faraday cage. We will assume a grounded large metal conductor at a quarter wavelength 'behind' the antennae. No waves will go behind the conductor. The waves do however have a specific wavelength this makes it possible to reduce the surface of the conductor to just 104% of the wave. As we have the antennae in a polarised position, we can also use this polarisation to reduce the size and shape of the reflector. A single rod behind the dipole will not only block any radiation back but also reflect forward. Two rods at 45 degrees up and down will even gain the signal another half dB.

Now we place more rods each 97% shorter in the forward direction these will guide the waves forward. (These should not be connected to whatever)

The used percentages are approximations as the wavelength in different metals will vary, I once build a 20 element Yagi with aluminium rods.