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What's the difference between machine zero and part zero?

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Machine Zero is the machine’s absolute coordinate system origin, unaffected by any kind of offsets.  On machines with home switches, this is where the machine goes when it is homed.

Part Zero is the coordinate system origin for the g-code.  It’s often convenient to have a particular feature of the part or of the workholding be part zero.  For example, on a lathe, we may want the finished faced surface of the part be Z = 0.  On a mill, we might want the left front corner of the fixed jaw of a vise be part zero.  

For more, see the chapter on Part Zero and Touchoffs from our G-Code Tutorial.

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