Eventually, we have a system that works like this (the simple version):

Design an object using 3D modeling software (possible output: STL file)
Convert 3D model into cutting instructions (possible output: G-code file or TPL (Tool Path Language))
Convert G-code into axis movement instructions (form?)
Convert axis movement instructions into motor signals (Arduino and motor controller)

How to Draw a Circle

Suppose we want to draw a circle of radius 3 centered at 5,5 on a 10,10 work area.

For simplicity, let's assume the pen or tool is already in the home or 0,0 position.
Go to 5,2 the "six o'clock" position on our circle.
Now we will cycle from y=2 to y=8 and back to y=2 with x being a function of y.