How the azimuth of the time base is synchronized with the azimuth of the rotating antenna?
Early radar sets used a pair Selsyn motors to transfer the rotary motion of the antenna platform to a sine-cosine potentiometer.

The terminals of the potentiometer were driven with complementary copies of the sawtooth waveform generated by the radar timebase, and the taps of the potentiometer fed the horizontal and vertical deflection amplifiers of the display tube, creating a radial scan at the correct angle that's synchronized to the receiver output.

Note that this system even allows the operator to interrupt the normal periodic scan in order to sweep the radar beam across a specific target. The display remains synchronized to the antenna at all times. Note also that "pan &amp; zoom" of the display is possible by varying the gain and DC offsets of the deflection amplifiers.

Modern digital systems simply use a shaft encoder on the antenna.