Mathematicians tell us that all curves can actually be broken down so small that for all intents and purposes they are made up of straight lines. Well, it is hard in the reality of life to get anything made of lines that small, so for a wood turner, three simple lines are all it takes to make the myriad of shapes that come off a lathe.
The cuts that are made are just a straight cut, a convex curve and a concave curve. These can used in a variety of ways to make almost anything that can be thought of for a lathe. Consider a table leg. It is generally made up of a square upper section, several beads and coves, a few decorative v-cuts and a curved foot.
The transition from pommel or square to the bead happens with a convex curve and then two more side by side but in opposite directions, the right side of the bead and then the left. Those decorative coves that lend a sense of lightness to the leg are a couple of concave cuts, one to the left and another to the right, meeting in the middle.
Many legs demonstrate a long “s” curve somewhere along their length. This is, from the top down, a concave cut right, a concave cut left, a convex right left and a convex right. The only thing separating it from a bead and cove is its length.
Flats are just a straight line and v cuts are simply two flats that meet at the bottom. All other cuts can now be seen to be mixtures of the three. Why, even a bowl can be seen to be a combination of an inside curve and and outside curve, then a flat along the bottom or perhaps another concave surface with a bead on the edge to give a rim to sit on.
So a turner needs to refine and practice those three cuts so as to have mastery in their execution. They will be needed over and over. The tools most used for the cuts are a skew for flats and convex cuts, and gouges for concave cuts. On the other hand gouges may be used for the former cuts as well as the latter. A skew chisel can be used for long cove or concave cuts but a gouge is generally deemed appropriate as it is used for those finer coves especially, where it is difficult to bring the skew to bear.
The challenge for the wood turner thus becomes two fold. First a design has to made using only the three elements and that design has to be clean and proportionate and, one would hope, unique. Then those elements need to become second nature in execution so as to leave a surface ready for limited sanding and a fine finish.