Hi Peter,
The video you linked works although it leaves a gap in the conical surface.
The sketch rip I described is the elegant solution.
A little while ago I helped an A level student who wanted aboriginal art on the outer surface of cylinder for wind chimes.It was tricky. We ended up with a tube in a tube. The outer tube was ripped in the method outlined. The flattened metal tube had the pattern applied then bent back. The inner tube was to se the depth of cut. The two assembled tubes were set up on a rotary axis on the mill for machining.
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Cheers
Tim
The trick is using a sketch line to rip the surface to allow it to flatten. I created a model which is attached.
- I revolved the shape then shelled, removing top and bottom.
- On the outer surface I created a tangential datum plane then sketch a line down the outside of the shape.
- The shape was converted to sheet metal model.
- A sketched rip selected the line which cut the cone.
- The flat pattern was applied resulting in a net for cutting.
So far tried lots of combinations of extrude with taper, sweep, swept blend (loft) and different geometries and either get non constant thickness error or the Flat Pattern stage fails. I'm pretty sure it's the variable radius that's causing the problem because a parallel extrude partially flattens like this:
I tried modifying the profile to give a flat surface for a surface rip.
Which worked quite well and would look "right" but got similar fail msgs to Tim
Nice 4axis mill by the waay!
Always happy to help, it keeps my brain active in advancing years. 
Looking forward to seeing the finished stool.
Peter using a round got me thinking and I managed to get a parallel sided shape to flatten in sheet metal but as soon as I added a draft it caused the flatten to create a weird partially curved shape. 
