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Default Calling all math wizards...

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?

Thanks...
--
Kevin Miller - http://www.alaska.net/~atftb
Juneau, Alaska
In a recent survey, 7 out of 10 hard drives preferred Linux
Registered Linux User No: 307357, http://linuxcounter.net

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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?

Thanks...


No math whizard here. Looks like the setup you show in the picture
would give all end grain showing, if that is any factor.

--
Gerald Ross

Everyone must row with the oars he has.






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Default Calling all math wizards...

Hi Kevin
I really do not see the advantage in cutting the segments that way Could you
please explain why it is done in that manner.

Looking at the picture it seems to me that as you true the circle up you end
up with a trapeziod any way.

Sorry I cant help with the math but I can give you the formulas for the
normal trapezoid segments. Either just as a mathematical formula or as a
formula for a spreadsheet.

Tom


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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
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On 12/11/2011 10:13 AM, Dan Coby wrote:
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)

.... snip

I need to add a correction to my last posting: IGNORE THE SECOND SET OF
EQUATIONS

I did not consider the case when the outside radius OR is only a little
larger than the inside radius IR. I suspect that this is commonly the case
for most segmented turnings. In this case you need to use the first set of
equations (which I called an error and wasteful). What can I say, it was
early in the morning here on the west coast when I wrote that. I am not a
morning person. A shower helped to clear my mind.

Your example picture has a large difference between the two radii. As a
result, the segments ware very wide and there is a large joint length.
My second set of equations works fine for this case.

With narrower segments (i.e. a small difference between IR and OR) we have
to make sure that when we get away from the joints that we still have
enough width to get the desired outside radius. The first set of equations
guarantees that.

A more accurate analysis would say the there are situations in which each
set of equations is better to be used. However, unless you indicate a
burning need, I am not going to bother to determine the cross over criteria.


Dan




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On 12/11/2011 04:58 AM, Tom wrote:
Hi Kevin
I really do not see the advantage in cutting the segments that way Could you
please explain why it is done in that manner.

Looking at the picture it seems to me that as you true the circle up you end
up with a trapeziod any way.


No, not so much. I'll post some photos in a minute that show what you
end up with.

Sorry I cant help with the math but I can give you the formulas for the
normal trapezoid segments. Either just as a mathematical formula or as a
formula for a spreadsheet.


I don't know the formula - there's so many places online that have a
working calculator that I've never bothered to look it up. For
instance, see:
http://www.delorie.com/wood/

Probably wouldn't be hard to find, but I figure why reinvent the wheel?

Google will turn up many more.

...Kevin
--
Kevin Miller - http://www.alaska.net/~atftb
Juneau, Alaska
In a recent survey, 7 out of 10 hard drives preferred Linux
Registered Linux User No: 307357, http://linuxcounter.net
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On 12/11/2011 04:52 AM, Gerald Ross wrote:

No math whizard here. Looks like the setup you show in the picture
would give all end grain showing, if that is any factor.


Now that's an interesting thing. Another poster also noted the same
thing. I didn't think he was right until I took a look at the glue-up.

Here's two photos using the reverse segment construction:

One is a bowl that I just finished a couple days ago. The second is the
glue-up I posted yesterday, but after I've trimmed it on the bandsaw and
lathe. On the bowl, you can see the end grain is the joint and not
really exposed. On the glue-up (which is to become the lid for the
bowl) it is pretty much end grain that is exposed.

I think one can see the difference between a normal trapezoidal ring and
the reverse ring quite easily in the photos though.

I wasn't really paying attention to the direction of the grain when I
glued it up. I didn't consciously do it differently than when I did the
bowl but apparently one can go about it two different ways. I'll have
to cut some more segments and experiment.

All part of the learning process...

....Kevin
--
Kevin Miller - http://www.alaska.net/~atftb
Juneau, Alaska
In a recent survey, 7 out of 10 hard drives preferred Linux
Registered Linux User No: 307357, http://linuxcounter.net

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On 12/11/2011 10:26 AM, Dan Coby wrote:
On 12/11/2011 10:13 AM, Dan Coby wrote:
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)

... snip

I need to add a correction to my last posting: IGNORE THE SECOND SET OF
EQUATIONS

I did not consider the case when the outside radius OR is only a little
larger than the inside radius IR. I suspect that this is commonly the case
for most segmented turnings. In this case you need to use the first set of
equations (which I called an error and wasteful). What can I say, it was
early in the morning here on the west coast when I wrote that. I am not a
morning person. A shower helped to clear my mind.


In that situation I'd probably use a traditional segment. The selling
point of the reverse segment method is the joints are angled rather than
pointing towards the center of the piece, giving a pin-wheel effect.
That is most dramatic on a wide segment and best viewed from the top
(see my posts today for an example). The effect would be pretty much
lost on a half inch wide piece. On a lid or a closed form however which
utilizes a wider 'face', your more accurate method will be a definite plus.

With or without a shower, it's all a muddle to me. Sure glad there's
further. That should givfolks like you that grasp this stuff!


Your example picture has a large difference between the two radii. As a
result, the segments ware very wide and there is a large joint length.
My second set of equations works fine for this case.


The piece I showed will be the lid to a bowl I made, thus needed to be
wide. It will have a solid cherry center and probably a finial of
whatever dark wood I can find laying around the shop.


With narrower segments (i.e. a small difference between IR and OR) we have
to make sure that when we get away from the joints that we still have
enough width to get the desired outside radius. The first set of equations
guarantees that.

A more accurate analysis would say the there are situations in which each
set of equations is better to be used. However, unless you indicate a
burning need, I am not going to bother to determine the cross over
criteria.


What you've provided is great. The amount of wood wasted using the
least optimal formula is pretty negligible as it is so no need to
optimize further. That should give me enough to play with quite
handily. I assume that any modern spreadsheet ought to have sin, and
cosine type functions so I should be able to plug in the values w/o much
trouble.

Thanks much Dan! Appreciate the input...

....Kevin
--
Kevin Miller - http://www.alaska.net/~atftb
Juneau, Alaska
In a recent survey, 7 out of 10 hard drives preferred Linux
Registered Linux User No: 307357, http://linuxcounter.net
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On 12/11/2011 10:32 PM, Kevin Miller wrote:
With narrower segments (i.e. a small difference between IR and OR) we have
to make sure that when we get away from the joints that we still have
enough width to get the desired outside radius. The first set of equations
guarantees that.

A more accurate analysis would say the there are situations in which each
set of equations is better to be used. However, unless you indicate a
burning need, I am not going to bother to determine the cross over
criteria.


What you've provided is great. The amount of wood wasted using the least optimal formula is pretty negligible as it is so no need to optimize further. That should give me enough to
play with quite handily. I assume that any modern spreadsheet ought to have sin, and cosine type functions so I should be able to plug in the values w/o much trouble.

Thanks much Dan! Appreciate the input...

...Kevin


You are welcome.

I did work out the criteria for when you need to use the first set of
equations.

If e*cos(theta) b then use the first of equations otherwise use the
second set of equations.


Dan

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On 12/12/2011 07:19 AM, Dan Coby wrote:

You are welcome.

I did work out the criteria for when you need to use the first set of
equations.

If e*cos(theta) b then use the first of equations otherwise use the
second set of equations.


Cool, thanks again.

FWIW, DJ Delorie created an online calculator that uses the equations at
http://www.delorie.com/wood/revseg.html

Appreciate all the help from everyone!

....Kevin
--
Kevin Miller - http://www.alaska.net/~atftb
Juneau, Alaska
In a recent survey, 7 out of 10 hard drives preferred Linux
Registered Linux User No: 307357, http://linuxcounter.net


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On 12/12/2011 7:13 PM, Kevin Miller wrote:
On 12/12/2011 07:19 AM, Dan Coby wrote:

You are welcome.

I did work out the criteria for when you need to use the first set of
equations.

If e*cos(theta) b then use the first of equations otherwise use the
second set of equations.


Cool, thanks again.

FWIW, DJ Delorie created an online calculator that uses the equations at http://www.delorie.com/wood/revseg.html

Appreciate all the help from everyone!

...Kevin


Great. I see that he implemented both sets of equations along with the test
for when to use which set. (And they appear to work. It is always nice to
see things when they work.)

I see two red circles and two blue circles. The red circles are the inner and
outer radii. One of the blue circles shows the inner edge of the segments. However
the purpose of the other blue circle is a mystery to me.

What is the purpose of the second blue circle?


Dan
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On 12/13/2011 09:13 AM, Dan Coby wrote:

Great. I see that he implemented both sets of equations along with the test
for when to use which set. (And they appear to work. It is always nice to
see things when they work.)

I see two red circles and two blue circles. The red circles are the
inner and
outer radii. One of the blue circles shows the inner edge of the
segments. However
the purpose of the other blue circle is a mystery to me.

What is the purpose of the second blue circle?


I don't know - I'll ask over in rec.crafts.woodturning. That's where DJ
posted is link and code to produce the output.


....Kevin
--
Kevin Miller
Juneau, Alaska
http://www.alaska.net/~atftb
"In the history of the world, no one has ever washed a rented car."
- Lawrence Summers
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On 12/13/2011 10:55 AM, Kevin Miller wrote:
On 12/13/2011 09:13 AM, Dan Coby wrote:

Great. I see that he implemented both sets of equations along with the test
for when to use which set. (And they appear to work. It is always nice to
see things when they work.)

I see two red circles and two blue circles. The red circles are the
inner and
outer radii. One of the blue circles shows the inner edge of the
segments. However
the purpose of the other blue circle is a mystery to me.

What is the purpose of the second blue circle?


I don't know - I'll ask over in rec.crafts.woodturning. That's where DJ posted is link and code to produce the output.


Thanks. I took a look at the postings on r.c.woodturning. From DJ's
equations I see that there are additional issues with N = 3 that I had
not considered.


Dan
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Dan Coby writes:
I see two red circles and two blue circles. The red circles are the inner and
outer radii. One of the blue circles shows the inner edge of the segments. However
the purpose of the other blue circle is a mystery to me.

What is the purpose of the second blue circle?


The inner blue circle is for the clamping brace. The outer blue circle
is the cutoff for when you switch to the other set of equations - it
identifies the size where you start getting end grain issues, too. When
the OD matches that blue circle, the OD circle is tangent to the edge of
the segment right at the intersection with the adacent segment.
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Dan Coby writes:
Thanks. I took a look at the postings on r.c.woodturning. From DJ's
equations I see that there are additional issues with N = 3 that I had
not considered.


At N=3 the angles are all leaning the "other way" so the sin/cos stuff
doesn't work out, plus you measure off the other side of each segment.
I tested N=2 also but there were infinities. I suppose if you can't
figure out the math for N=2, you probably shouldn't be using heavy
equipment with sharp objects.


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On 12/14/2011 1:25 PM, DJ Delorie wrote:
Dan writes:
I see two red circles and two blue circles. The red circles are the inner and
outer radii. One of the blue circles shows the inner edge of the segments. However
the purpose of the other blue circle is a mystery to me.

What is the purpose of the second blue circle?


The inner blue circle is for the clamping brace. The outer blue circle
is the cutoff for when you switch to the other set of equations - it
identifies the size where you start getting end grain issues, too. When
the OD matches that blue circle, the OD circle is tangent to the edge of
the segment right at the intersection with the adacent segment.


Thank you for the information. I had noticed that as I change ratio of the
sizes of the outer and the inner radii, that the 'mysterious' blue circle
did converge onto the outer radius when we hit the crossover point between
the 'thick' and 'thin' segment logic. However I had not made the connection
about 'cutting uphill' when turning. (I am not a turner.)

I have been enjoying playing with your calculator so I would also like to
thank you for your efforts in creating it and making available to others.


Dan
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On 12/14/2011 1:27 PM, DJ Delorie wrote:

Dan writes:
Thanks. I took a look at the postings on r.c.woodturning. From DJ's
equations I see that there are additional issues with N = 3 that I had
not considered.


At N=3 the angles are all leaning the "other way" so the sin/cos stuff
doesn't work out, plus you measure off the other side of each segment.


Yes. When I first looked at your display for the N=3 case, I wondered why
you had reversed the orientation of the segments. Then I realized that
you really had not changed the orientation, instead the angle had swung
past 90 degrees and that appears to change the orientation. (And, as you
said, that also changes which edge of the segment is the 'length'.)
Thanks for adding to my education.


I tested N=2 also but there were infinities. I suppose if you can't
figure out the math for N=2, you probably shouldn't be using heavy
equipment with sharp objects.


Yes. :-)



Dan

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On 12/14/2011 02:20 PM, Dan Coby wrote:
On 12/14/2011 1:25 PM, DJ Delorie wrote:
Dan writes:
I see two red circles and two blue circles. The red circles are the
inner and
outer radii. One of the blue circles shows the inner edge of the
segments. However
the purpose of the other blue circle is a mystery to me.

What is the purpose of the second blue circle?


The inner blue circle is for the clamping brace. The outer blue circle
is the cutoff for when you switch to the other set of equations - it
identifies the size where you start getting end grain issues, too. When
the OD matches that blue circle, the OD circle is tangent to the edge of
the segment right at the intersection with the adacent segment.


Thank you for the information. I had noticed that as I change ratio of the
sizes of the outer and the inner radii, that the 'mysterious' blue circle
did converge onto the outer radius when we hit the crossover point between
the 'thick' and 'thin' segment logic. However I had not made the connection
about 'cutting uphill' when turning. (I am not a turner.)

I have been enjoying playing with your calculator so I would also like to
thank you for your efforts in creating it and making available to others.


Many thanks to both you and DJ.

Cutting uphill refers to cutting in a direction that lifts the wood
fibers from the piece. I assume you are a wood worker, so it's
essentially the same thing as running flat stock through a jointer or
planer with the grain facing the wrong direction.

When turning a piece with the grain parallel to the lathe bed, you want
to cut from the widest part inward towards the center. That way the
wood fibers are supported by the wood underneath them as you cut. When
doing bowl turning where the grain is perpendicular to the lathe bed you
do the opposite - you cut from the center area outwards...

....Kevin
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Kevin Miller - http://www.alaska.net/~atftb
Juneau, Alaska
In a recent survey, 7 out of 10 hard drives preferred Linux
Registered Linux User No: 307357, http://linuxcounter.net
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Default Calling all math wizards...

On 12/15/2011 6:47 PM, Kevin Miller wrote:
On 12/14/2011 02:20 PM, Dan Coby wrote:
On 12/14/2011 1:25 PM, DJ Delorie wrote:
Dan writes:
I see two red circles and two blue circles. The red circles are the
inner and
outer radii. One of the blue circles shows the inner edge of the
segments. However
the purpose of the other blue circle is a mystery to me.

What is the purpose of the second blue circle?

The inner blue circle is for the clamping brace. The outer blue circle
is the cutoff for when you switch to the other set of equations - it
identifies the size where you start getting end grain issues, too. When
the OD matches that blue circle, the OD circle is tangent to the edge of
the segment right at the intersection with the adacent segment.


Thank you for the information. I had noticed that as I change ratio of the
sizes of the outer and the inner radii, that the 'mysterious' blue circle
did converge onto the outer radius when we hit the crossover point between
the 'thick' and 'thin' segment logic. However I had not made the connection
about 'cutting uphill' when turning. (I am not a turner.)

I have been enjoying playing with your calculator so I would also like to
thank you for your efforts in creating it and making available to others.


Many thanks to both you and DJ.

Cutting uphill refers to cutting in a direction that lifts the wood fibers from the piece. I assume you are a wood worker, so it's essentially the same thing as running flat stock
through a jointer or planer with the grain facing the wrong direction.

When turning a piece with the grain parallel to the lathe bed, you want to cut from the widest part inward towards the center. That way the wood fibers are supported by the wood
underneath them as you cut. When doing bowl turning where the grain is perpendicular to the lathe bed you do the opposite - you cut from the center area outwards...


Yes. I understood what DJ meant by 'cutting uphill' why it is a situation
to be avoided. I had not made the association between the blue line and
cutting uphill until DJ explained the purpose of the line.

Dan

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