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Cartesian printers move one or two motors along each of the X, Y, and Z axes and the name was derived from the Cartesian coordinates system. They typically have a rectangular build area and the printers themselves tend to have a cube-like shape. The Lulzbot Mini is a fine example of these types of printers.
Delta printers have three arms that come together in the center to suspend the extruder above the build area. Deltas also use a Cartesian coordinates system to move around in, but instead of moving one motor per axis at a time, all three arms move at different rates or times to precisely move the nozzle with triangulation. The SeeMeCNC Rostock MAX V2 is a prime example of a delta printer.
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These chips are responsible for running the stepper motors. They fire the coils of the motor in sequence, causing it to move in increments. Many motherboards have the stepper drivers built in, but some also have them in modules that can be unplugged. By balancing the power fed to each coil, the driver is also able to divide steps up into further increments. This is called microstepping, and allows more precise control over the motor than is normally possible. The stepper driver also controls how much electrical current is fed to the motor. More power makes the motor stronger, but also makes it run hotter. See this article for more information on adjusting your motor current.
A set of stepper motor drivers
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Some printers have an LCD screen so they can be controlled directly without hooking them up to a computer. These can be basic black and white displays like the VIKI 2 or advanced wifi enabled touchscreens like the MatterControl Touch.
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MatterControl T10
SD Card Slot
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Many printers have some kind of a system for automatically making sure that the bed is level with the nozzle. Some do not, though, and must be calibrated by hand. MatterControl also has the ability to account for unlevelness in software. For more information, see our wiki article on bed leveling.