How do you control a polar 3-D printer?

Like, you have a turntable with your workpiece on it, and another
turntable that moves your tool, that can swing it over the center of
the first turntable.  And you can move your other turntable up and
down.  Now how do you convert x and y coordinates into angular
positions?

Perhaps a handy thing to know is that the polar equation r = sin θ
produces a circle running through the origin, retracing the same
circle when θ is in [0, π] and in [π, 2π], with its center at r = ½, θ
= ½π.  So, given the radius r at which a point is located on the
workpiece turntable, you can rotate the workpiece turntable to any θ
such that sin θ = r, such as sin⁻¹ θ, to give the tool turntable a
chance to hit the point.  Then it’s just a matter of finding the angle
for the tool turntable, a simple atan2.

In short, starting from x, y on the workpiece turntable, scaled so
that the distance between turntable centers is 1:

    θw = sin⁻¹ √(x² + y²)
    s = sin θw
    c = cos θw
    θt = atan2(1 - (c·x - s·y), s·x + c·y)

If you’re trying to design a toolpath, it may be worthwhile to take
advantage of the other alternatives available for θw.