This article shows one way to make the crescent-shaped piece from a piece of wood that has near-correct natural curvature. This particular piece of wood is from root of a spruce. There are also other ways to make this component.
Splitting the piece Edit
Attach the piece to a vise:
Plane the top and the bottom flat with a jointer plane:
Mark the curvature of the crescent-shaped piece, preferably to the side that ended up being less straight and smooth. If you happen to have a bandsaw, then you can simply split the piece along the line you drew. If not, make a series of indentations along the line with a bradawl:
Drill a series of holes (~8mm) through the piece using a bench drill; this ensures that the holes are exactly vertical:
If the drilling depth is not sufficient, finish the holes with a hand drill while the piece is attached to a vise:
The other side of the piece will look a bit rough at this point...
...so you can smoothen it with a plane:
Next saw slits to one end of the piece to start a split:
Beat a small axe into the slit, while the piece is attached to a vise:
You should start hearing cracking noises as you force the axe blade in.
Saw another slit to the other end of the piece (this could also have been done earlier):
Attach the piece into a vise the other way around:
If your piece does not have twisted grain and is in otherwise perfect shape, you should get a clean split. If necessary, assist the split by sawing or chiseling away wood from between the holes. The end result will look crude, but we're not interested in aesthetics at this point:
However, in this particular case there was some twist in the grain and some damage from micro-organisms, and the outer side cracked in the process:
Still, more than half of the outer side ended up being usable.