Hi Dennis, I see what you mean.
I was thinking of taking off a short, thin shaving from near the end, turn a very small amount and remove a second shaving just like the first. Go all round like that then start taking off the second "round" of shavings starting further from the end after each lap.
How about a simple V groove to get down to a round dowel shape first, or are you starting with round dowels and just putting a taper on one end( or both ends) afterwards?
If you start with round wood on the tapered V groove, you would remove a little wood with one shaving and turn a few degrees before taking the next shaving off. The shavings would be short to start off with, like sharpening a brand new pencil takes off a narrow shaving from the end mm or so , all round before your next turn takes off a wider shaving. By the time the wood has turned 180 degrees , only a short piece at the end has been removed. Most of the length is still pressed against the V groove and is supported. If the tapered V groove shooting board is only as long as the taper you want to form, there will be no reason for the tapered end to become unsupported. Again, like the pencil in the sharpener, it fits into the taper better and better as the work progresses. I realise the pencil sharpener cuts in a different direction, but I think narrow shavings produced by small rotations will give similar results.
If the taper V groove is as long as the taper you want on the arrow, as your work progresses and the shavings get longer, so the back end of the arrow is lowered getting a longer contact between the taper length you have cut and taper support from the V groove.
That method could be used in reverse. Use the long taper groove but just rest the tip of the arrow at the wide end of the taper groove. Take off shavings all round near the end. Next time round, hold the shaft a little further into the V groove towards the narrow end. Take off more shavings all round and as each lap is made, put the shaft further towards the narrow end each time. It sounds complicated but my guess is that the length of the shavings will be about the same as the length of the shaft held onto the V groove. So the plane will start cutting at about the wide end of the V groove. I appreciate the wood may move a little along the groove with nothing to stop it, but you will be holding the other end of the shaft and with a sharp plane, the tendency to move will be slight. Remember the shavings may be getting longer but they will always be narrow ones, so not much wood is being removed at any one time.