kö-ko-gen and other methods

For those wishing to discuss the material found in The Art of Japanese Carpentry Drawing essay series.
roberts
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kö-ko-gen and other methods

Tue Aug 09, 2011 4:48 pm

Perhaps it is my fault, but I am confused.

Over the last few days I have been comparing the kõ-ko-gen method for obtaining hopper angles with other methods.

a) The chõ-gen of a 10/4.9 kõ-ko-gen triangle is 8.9899
Using this value in "sashigane-mode" gives an angle of atan(8.9899/10) = 41.923°

b) If you consider the corner of our hopper as one corner of an inverted pyramid then the standard pyramid formula apply.
For a rise/run, opposite/adjacent, kõ/ko of 10/4.9 the dihedral angle between two adjacent faces is 101.164°
Halving this and taking its complement (a sort of woodworkers fix) gives (90- (101.164/2)) = 39.148° .

a) 41.923° versus b) 39.148° - a considerable difference.

I think this is an example of the difference between a) marking the miter cut on the board and cutting by hand, and b) setting up a power saw and running through a horizontal board.

Is there an error in my reasoning ?
If it is correct, what is the relationship between the angles a) and b) ?

Help would be much appreciated .
Rob
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Chris Hall
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Re: kö-ko-gen and other methods

Tue Aug 09, 2011 8:59 pm

Your reasoning is correct in regards to the different angles, however the chōgen angle on the edge of the board, taking 10 on the long arm of the square and 8.9899 on the short arm is (opposite/adjacent) 10 / 8.9899 = 1.1237375. Take arctan to produce the miter angle of 48.076˚. You might note member CraigAK's questions about the chōgen angle raised in earlier discussion, titled 'Framing Square and the Chōgen':

http://www.thecarpentryway.com/Carpentr ... p=296#p296

Keep in mind that was from TAJCD Volume II, and we were discussing a 10 / 4 unit triangle.

Since the dihedral is taken at 90˚to the board face's miter cut (the chūkō miter line), this angle cannot be the same as the chōgen angle since the chōgen angle is laid out off the board edge. The table saw makes the cut along the line of the face's miter cut, so this by default is 90˚ to that miter line.

If you follow the same type of triangle breakdown I used in Mailing 2, pages 10 onwards you will be able to compare these angles and see how they relate.

~C
roberts
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Re: kö-ko-gen and other methods

Wed Aug 10, 2011 9:36 am

I was very afraid of getting this post wrong - and I did, in a way.
Angles and their compliments is a constant source of banana skins. Even if you calculate the miter angle as 48.076, the chances are that that the scales on your saw, and its tilt direction, expect you to compliment it.
I am part way through making the Hopper for the TAJCD exam requirement - angles and their compliments is one of my main worries. You need a very clear head and a lot of similar timber to test things on (I have just done a test and got the butt-miter going the wrong way. It produced a very sharp right-angle at the "eaves" !)

Thanks for the leads, Chris, I will go back and look at them again.
Rob
roberts
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Re: kö-ko-gen and other methods

Sun Aug 14, 2011 11:28 am

Subject could be "kö-ko-gen outside Japan"

While scanning the Internet for ways to set out hopper miters I came across the book Constructive Carpentry by Charles A. King, published in the USA in 1912:
http://www.archive.org/details/construc ... 00kingrich
The pages 94 to 97 describe the layout of purlins which, as the text states, are closely related to hoppers.
To my surprise, Fig 89 on page 96 shows a kõ-ko-gen triangle, although of course it is not called that.
Notice that King uses (chûkõ / tan-gen, his A / R) as (rise / run), which by similar triangles, is the same as kõ/ko.
Very interesting.

Footnote: Even after the full explanation in the topic CSG Mailing 5, I am not confident that I have got kõ and ko the right way around.
Is this example about roofs (purlins, low angle of interest) or hoppers (miters, high angle of interest) ?
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Chris Hall
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Re: kö-ko-gen and other methods

Sun Aug 14, 2011 3:56 pm

Robert,

if you look at page 97 of that text, fig. 91 shows the use of chūkō, by way of graphical development for obtaining the face cut of the board.

As far as a fuller understanding of ko and , I have made fairly extended explanations to this point, and suspect a certain amount of time is sometimes required for things to 'sink in'. That said, I suspect if you wait until we are working with a roof model and I delve into the kōkogen method in that application, the picture should fill out for you a bit.

~C
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Re: kö-ko-gen and other methods

Mon Aug 15, 2011 2:02 pm

Yep, I had found the "fig 91" method in an even older text: "Modern Carpentry and Building" by W.A.Sylvester, published in 1896 ! Also on http://www.archive.org .
I had not interpreted that method with the chūkō in mind. I had used the figures (33, 34 in this book) to go through the trig. Must look at them again with different "glasses" on.
Robert

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