This has been a really fascinating discussion.
Evil Me ponders if anyone ever proposed a radial raster (like the classic PPI RADAR displays)
While the traditional way to produce these involved mechanically-rotating the scan coils round the neck of the CRT (synchronised using Selsyns etc. to the movement of the scanning-antenna) I will admit that a few years back for a computer-games business I 'reverse-engineered' a Radar-type PPI-display presentation for one of their products.
After a few false-starts I realised that rather than gobbling realtime compute-cycles for each-and-every sine/cosine point of each possible 360-degree rotation (they were expecting at least 30Hz refresh), it was easier and display-faster to malloc() half-a-Megabyte of memory, do a single-pass vector-points computation (which took a second or so, easily lost in loading-time) and thereafter treat the whole thing as a simple lookup-table for X and Y values.
PPI-style TV could have been 'interesting'...