This is pretty sweet, how he turned these first into wooden travel mugs, then laser engraved them. I know we won’t be able to do something like this on the GF due to the lack of a rotary axis, but pretty cool anyways. Of course depending on how good the depth focus works, we might be able to do it by removing the tray?
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assuming it works as they suggest, you should definitely be able to do a narrow vertical band; your patience and alignment skills will determine whether you could do this piecemeal 
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As @Kusmeroglu showed in the stained glass project, the first thing would be to make an alignment jig, then do this. With a jig this would be quite repeatable.
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sure, i meant aligned in terms of lining up each engraving on the cup. i’m skeptical it’ll look as good as being able to do it in one go.
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If not piecemeal then maybe as a flat, then meet bend maybe?
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oh yeah, absolutely.
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How about lasering a veneer and wrapping it around a travel mug to customize it. No turning needed and it fits in a GF. I could do that pretty quickly and the result would be indistinguishable from the link. - Rich
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Some of that Proofgrade veneer with 3M adhesive that has been talked about could work well.
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yeah, that’s what we were just talking about. pretty easy option for straight-sided vessels. bit more complicated on a slant.
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That is probably the only thing that will allow this, as most coffee cups are about 3" in diameter.
Edit ) I suppose one could be cut in half and lasered, then glued back together.
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A conic surface shouldn’t be a problem - think paper lamp shades.
Fitting text to a curve is likewise, available in most graphic software.
The tricky part comes with trying to map a picture to a curve, and I’ve spent many hours trying with early CorelDraw, version 8.
If anyone can point me to later software that does it, or alternative routes, I’d be grateful.
Your friendly fan maker, 
John
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i didn’t say it would be impossible, merely trickier.
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Gotta make me some of these!
Having a new wood lathe and all
Awesomeness 
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The veneer will work as a covering for sure. If using the turned piece, just do a recess carve the thickness of the veneer and then glue it on. Lots of ways the veneer could be used.
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Oh, that’s a nice idea.
I’d like to do something like this!
Beautiful!
I believe that “Make the Cut” software has that built in. (maping a graphic- not just text to a conical wrap)