But getting equal time on each pixel as you spiral out would be crucial to trigger the right phosphorescent equal luminance from your light-emitting material. Theta would not be linear with time.
The intensity of the beam could be modulated to counter this.
There are sync pulses on every line of analog video, as well as to color burst in most analog video standards. You couldn't just sweep theta and r continuously for a whole frame.
Why not? The sync pulses are because the scan is discontinuous, you could just have one plus color burst every spiral
I'm guessing that NTSC and PAL have a color burst on every line, vs every frame, because over an entire frame the local oscillator might not be stable enough. At least back in the day.
Same with the timing of the sweep circuits, which only need to hold sync for a single line of video.
x = r cos (theta) y = r sin (theta)
when you vary r so it gives you a spiral instead of a circle. So it shouldn't be too hard to generate the signals required.
Or there could be a different arrangement of the beam deflectors that didn't give you independent control of X and y.