On 12/10/2011 8:11 PM, Kevin Miller wrote:
The usual way to do segmented turnings is to cut trapezoids and glue them up in a ring then stack the rings up to make the vessel. Typically, one cuts the same angle at both ends
of the segment.

When doing reverse segments, one cuts one end to the appropriate angle, and leaves the other end at 90 degrees. The angled edge of the segment is oriented outwards rather than
inwards. See my facebook post for an example of a ring being glued up:
https://www.facebook.com/media/set/?...l=f 7a342b1e7

When cutting normal segments, one sizes them for a given outside and inside diameter. There's lots of segment calculators on the web that will give you the width of the board to
use, and the length of each segment. However I can't find any calculators that will determine the dimensions for a reverse segment except I have a spreadsheet that calculates the
width and length for an eight sided ring using the following formula:

OR = outside radius
IR = inside radius

Width = OR - (.924* IR)
Length = (.541*IR+Width)/.707

The segment length seems to come out a little long but that gives some fudge factor so that's fine.

What I'm looking for is the formula to enter the number of segments, the inside and outside diameter and for it to calculate the length of the long edge of the segment. Any math
whiz out there that can clue me in?

As you said the equations that you show give a result that is a little long.
This is because they calculate the width (and from it the length) a little
long. They calculate the width measuring the OR perpendicular to an edge
of a segment. A more accurate (and less wasteful) value uses a diagonal.
The error in their method will get worse as you increase the number of
segments.


Making the same error that they did:

theta (the angle) = 360 degrees / number of segments

width = OR - cos(theta/2) * IR

length = 2*sin(theta/2)*IR + width/sin(theta)

or length = (2*sin(theta/2)*sin(theta)*IR + width) / sin(theta)

Check for number of segments = 8:
theta = 45 degrees
theta/2 = 22.5 degrees
cos(theta/2) = .9238 (check)
sin(theta/2) = .38268
sin(theta) = .7071 (check)
2*sin(theta/2)*sin(theta) = 2*.38268*.7071 = .5411 (check)


The more accurate version:

theta (the angle) = 360 degrees / number of segments

b = IR * sin(theta/2)

c = IR * cos(theta/2)

d = square root(OR**2 - c**2)

length = d + b

e = d - b

width = e * sin(theta)


Dan