Segmented turning can be confusing in the beginning, particularly the part of calculating the miter cuts. Some people shy away from it because they don’t want to deal with the math. The method that I use is different than how it is described in most books. I will attempt to describe how I calculate the cuts.
Fig1 shows the conceptual sketch. This does not have to be perfect. This is a segmented vase with nine 3/4" thick rings. Each ring will have eight segments. That means 22.5 degree cuts. You may have noticed that in Fig1 there is a tenth bottom ring as well. I have corrected that in Fig2 by adjusting my design though. Fig2 is the half profile with the added inside wall.
The vertical lines define the outer wall and inner wall edge for each ring. Point A, Point B and Point C are different for each ring. I have only shown these points for ring 1 though.
In Fig3 I have split each 90 degree quadrant in half so that each 45 degrees can be used to draw one segment. Since I have 9 rings but only eight 45 degree sections, I will use one of the sections to draw both ring 2 segment and ring 9 segment.
Use the two hands of a compass to transfer the distance between Point A and Point B from Fig2 onto Fig3 and create a 45 degree arc. Then transfer the distance between Point A and Point C. Do this for each ring.
Ring 9 will only contain one arc, as this is the bottom and it has no inner wall.
Draw a line for the ‘Outer Length’ and ‘Inner Length’ of the segment. The outer length and the inner length lines were drawn 1/8” to ¼” away from the curves. This is to allow a cushion. You can set it to whatever you are comfortable with.
My intention here was to give an alternate method that I find simple to follow. If you find it even more confusing then please accept my sincere apology.