This is an 'cost down' version of 3D movement that is intended for drilling PCBs via CNC but it could as easily be used as a 3D printer by adding a hot end and extruder.
First up, I will pay some homage to the project linked below as it really was the inspiration -
Most of the math is here -
But I will have to redo most of the math so that it can work with integers on an 8-bit micro. I am currenly using a RAMPS 1.4 shield ontop of an Arduino Mega (ATmega2560) 8-bitter.
Linear rails / bearings and ball screws / threaded rod are very expensive.
This alternative uses slightly more expensive steppers but uses no linear motion components, just cheap extruded aluminium instead.