OK, that might seem like an odd (and pretty wide) question.... so let me boil it down by way of a couple of examples...

Let's say I wanted to machine a straight line between the position my cutter is at now, and another position which is 10 inches away at an angle of 14.8 degrees (some nice random numbers there...)

I'm assuming that the controller figures out that for every N steps in the X-direction, there must be 1 step in the Y-direction. But how does it then apply this to the controller? Does it literally move N steps in the X-axis, then 1 step in the Y, repeat until we arrive at our destination? If N is not a whole number, presumably it keeps track of the remainder & occasionally moves 1 step in the Y axis at N-1 steps, or N+1, or whatever.

OK, so next question... This time we need to move in an arc. I can't remember exactly what my machine requires to machine an arc (in polar coords), but let's say it takes the distance around the arc, and the radius - and the start point, which is the current cutter position. I'm assuming(!) that the controller, in this case, "chops" the arc into hundreds or thousands of little straight lines, then applies the same logic as above (N X-steps, 1 Y-step, or 1-X step, N Y-steps, whatever), would that be correct?

So... my final question... but first, a bit of setup...

Imagine an upside-down "V" shaped pair of identical length arms. Linked together a bit like a pantograph, but in the vertical axis. As the angle between arm "A" and the bed changes from zero (flat) and 90 degrees (dead upright), the angle between arm "A" and arm "B" changes from 180 degrees to zero degrees. i.e. the joint between the arms is always at the exact mid-point of the distance between the arms.

With me so far? Then I shall continue...

Imagine arm "A" is mounted at the centre of a rotary table. It can be rotated between 0 and 180 degrees, where 0 points due south, and 180 points due north.

If the arms were 1 meter long, therefore, at maximum extension the far end of arm "B" would describe a semicircle some 2 meters in radius. Depending on how close in one could get, this gives us a reasonable working rectangle of 1.5 x 2.5 meters, with a little reserve left in the tank.

Anyway, assuming you've got this far.... does anyone have any idea of what the maths would be to convert that straight line, or (worse) the arc, into the 2x angles needed to drive such a contraption?

Postscript: This is the sort of thing that happens when you've got all day to yourself, and no workshop to play in... I've been following Andrew Mawson's CNC plasma adventures with much tool envy... but, with the best will in the world, I just don't have room for a flat-bed CNC plasma cutter, even a home made one. Well, unless I can hang it on the wall maybe... So. my hyperactive mind thinks: Robot arms... if you made it compact enough, in theory it would take up very little space when "parked", and could be more or less bolted to a regular bench when in use. Yes, you still need the space when it's actually being operated; but that can usually be arranged. For storage, it all folds up and out of the way... With this in mind, I figured if it were accurate to the nearest 1mm, that's probably close enough for plasma work; in fact, in the "distance" angle, a resolution of 0.01 degrees allows precision no worse than 0.3mm per step at the full 2m stretch. I haven't figured the rotation accuracy yet, but I'm assuming it'd be similar. This would be fairly easy to achieve using a 10:1 driver & 36-position rotary encoder. And beyond that... I haven't thought any further yet. Thought I ought to get some knowledge about how CNC systems actually work first. Hence, the question