If machining full circles only, would it invoke trigonometry?
Pythagoras perhaps – the only dimension involved is the radius and that would be a function of long and cross distances from the spindle axis. Trig. needs angles knowing, but I think here they are not needed.
I am thinking of machines with round column as well as square ones, and applicable whether working from dials or DRO.
If you start at the rotary-table and spindle on a mutual (0,0) position then the radius is the hypotenuse of a triangle formed by the two linear displacements. Work back from the radius and a sensible cross-travel to obtain the longitudinal travel.
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A few years ago I had a slightly similar problem when I needed drill sets of bolt-holes on pitch-circles on several plain discs. These had all been turned and the drilling was the second operation.
The radial accuracy was not too critical, within reason, as long I made the mating parts on the same set-up, and a rotary-table clamped down on a bench-drill was fine for the task. The holes accommodate only ordinary bolts.
I also realised the starting-point angularity did not matter, nor did centring the RT under the spindle, because the radius is always constant from the RT axis.
I marked out the discs, so could swing the head round the column until a centre-drill met the marked circles with the RT reading 0º, clamped all up, measured to verify the setting; then needed use only the degrees to locate the holes.
(I forget the details now but might have used the tool point in the lathe, as a scriber to mark the pitch-circle from the centre.)