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Definition of each lathe axis

The Spindle

The axis parallel to the spindle axis is always Z, whether its a mill or lathe. And, for the lathe, the axis that controls the diameter is always X.
 
And the Lathe rotary axis conform to the standard of A rotates around X, B around Y, and C around the Z.
C is the most common on a Lathe, as it relatively easy to turn the chuck into a C axis.
M.
 
Yes, Z and X are what I've heard the most of in the manual world. So Z and X are the standard in the CNC world, right?

It's called the "right hand rule". Orient the thumb, index and bird finger of your right hand 90° from each other. Point your thumb along the spindle centerline, in the work area of the machine. Your index finger is X, your bird finger Y and your thumb is Z. All fingers should point positive.
Axis A, B, C are relative to the three primary axis. A is rotational about the X axis, B is rotational about the Y axis, and C is rotational about the Z axis. (These can also be designated as rx, ry & rz, most notably on robots.)
Note: This does not _always_ apply, as there are some odd-ball and downright weird machines out there, mostly in specific applications.

Robots work via the left hand rule. Same deal, but with the left hand and relative to the base, not a spindle.
 
I think that would be the left hand rule. I have never seen a machine (lathe) where moving towards the spindle was a positive move.
On a mill up is positive, an dit would be a left hand rule.
C rotates about Z
A rotates about X
B rotates about Y

On a lathe moving the tool away from the centerline is positive and moving teh tool away from the spindle is positive.
 
Okay

Right hand should work in all cases.

Right hand, palm up, thumb to right, index away from you, middle finger straight up.

Thumb is X positive
Index is Y positive
Middle is Z positive

This applies to pen and paper, and milling machines

Move so that index finger is up. This applies to CAD/CAM systems, where Z plus is coming at you from the screen (default view, mostly.) X (thumb) is still pointed right.

Turn (contort) so that the index (still Y) is pointed down, middle is pointing right and thumb is towards you. This applies to lathes with the turret working from the operators side of the spindle.
This is why on these machines, G02 and G03 as well as TNRC seem backwards.


Contort some more. Index points up, middle points right and thumb points away from you. This applies to lathes that have the turret approaching the work from the side opposite the operator.
Here, G02 and G03 and TNRC seem correct.



Haven't had experience on HMC's so I'm not familiar with that configuration, but I'm sure it agrees with convention.
 
This applies to pen and paper, and milling machines

And Oscilloscopes. :D

Haven't had experience on HMC's so I'm not familiar with that configuration, but I'm sure it agrees with convention.

One thing to remember with Mills, the spindle is often fixed in the X/Y so the table moves in the 'opposite' direction to account for this.
M.
 








 
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