A barrel back is curved in two directions at once, and that is the difficulty: it curves in plan, wrapping around the sitter, and it also leans back as it rises.
Buttoning that surface is not the same problem as buttoning a flat panel. On a compound curve the distance between two button positions changes depending on which way you measure it, so a grid set out to equal spacings will come out visibly stretched at the outer edges of the barrel and compressed at the centre.
Our upholsterers set the grid out on the curved timber frame itself, walking a flexible rule over the actual surface rather than measuring across it, and mark each button where it will truly fall. The covering is then cut with graduated fullness so the diamonds between the buttons come out even to the eye across the whole barrel.
The gilded scroll that runs down each front edge is fitted last, after the buttoning is complete, so it lands exactly on the edge of the upholstery with no gap - which is only possible in that order.