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#1
Posted to alt.binaries.pictures.woodworking




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 
#2
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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. 
#3
Posted to alt.binaries.pictures.woodworking




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 
#4
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 
#5
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 
#6
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 
#7
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 glueup. 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 glueup 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 glueup (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 
#8
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 pinwheel 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 
#9
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 
#10
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 
#11
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 
#12
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 
#13
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 
#14
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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. 
#15
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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. 
#16
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 
#17
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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 
#18
Posted to alt.binaries.pictures.woodworking




Calling all math wizards...
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  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 
#19
Posted to alt.binaries.pictures.woodworking




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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