I have a LASER that is made to mount in a chuck or collet. It has alignment 
screws and adjustable focus. Aim it at a wall far away and adjust the focus for 
as small a dot as possible.
Make a target to mount on the carriage, chuck and center the body of the LASER 
then use the alignment screws and adjusting the target and rotating the spindle 
until the spot no longer describes a circle when the spindle is rotated.

Move the carriage to the right end as far as possible. Move the cross slide 
intil the dot is on the target. There's your zero. Advance the carriage toward 
the bed until the dot moves off the target. Adjust the carriage until the dot 
is back on target, at least in the X axis. A really good way to map the bed 
would be to have a sensor as the target so the carriage could be moved by the 
control and read a continuous stream of how far off X it gets. Barring that, 
stop every X distance then adjust to re-zero and interpolate slopes between 
adjustment points.
If you put the target on a vertical slide to re-zero Y at each stop, the error 
map (with appropriate math) could compensate for that too. Perfectly straight 
cutting even with a very worn bed, as long as the vertical error doesn't drop 
the cutting tools too low.

    On Monday, September 10, 2018, 1:55:36 AM MDT, Chris Albertson 
<albertson.ch...@gmail.com> wrote:  
 The lasers are easy to find on Amazon but be warned the laser is not
perfectly aligned with the housing.    But as I said, this does not matter
if you spin the spindle.

I think a laser is the only option.  If you are measuring the bed your
reference needs to be at least 10X straighter then the bed.  Precsion rods
are not good enough but a $20 laser is literally perfect (as long as you
rotate the spindle.)

With effort you can use the same laser to measure the error with the lead
screw pitch. Use it as a laser range finder or laser inferometer

But all this measurement may be moot if the lathe is not reputable.  The
error might be random.  Maybe the carriage moves like a tuck on a dirt road?  
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