Hogen

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Chris Hall
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Hogen

Wed Dec 09, 2015 10:56 pm

I was delighted the other day to finally figure out a graphical way to show how the hogen slope is derived. This was something relatively easy to do with the face and miter cuts of a hopper, but hogen/shō-chūkō were another matter. Probably will be amending Volume II in the near future to show this.

Also have been studying splayed post layout in a little more depth of late, as it forms part of the cabinets I'm working on. Anybody else working on compound joinery these days?
Jack_Ervin.
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Re: Hōgen

Thu Dec 10, 2015 1:31 am

Chris,
I've been following along with your current build and am interested in what you have discovered about how the hōgen slope is derived.
Jack
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Chris Hall
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Re: Hōgen

Thu Dec 10, 2015 8:17 pm

I'll post a drawing up in the near future.
Jack_Ervin.
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Re: Hōgen

Thu Dec 10, 2015 8:40 pm

Chris Hall wrote:I'll post a drawing up in the near future.
I have a small step stool in the works with legs splayed 10/2. Sticks rough cut and drawing ¾ done. I'm in no hurry since I know you have much on your plate of major importance and I have plenty of other duties to attend to.
Thanks,
Jack
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Re: Hogen

Sat Jan 30, 2016 1:17 pm

Took a while longer to get around to this - my apologies.

Here's the basic situation where laying out the hogen slope will produce a return (lip) on one of the pieces meeting at a hopper corner:
Hogen Drawing.jpg
Hogen Drawing.jpg (61.37 KiB) Viewed 6544 times
Now, we can represent the intersection of the parts with an elevation view:
hogen 5.jpg
hogen 5.jpg (173.69 KiB) Viewed 6544 times
Since it is an elevation view, and the parts we are joining are at slope, we cannot use the elevation view to obtain our cut angles or actual lengths of parts for the facing board. The adjoining board, of which we see an end grain print in the elevation view, is depicted at actual x-section size of course.

In the elevation view, we can ascribe our unit triangle to the view of the adjoining board end, as follows:
hogen 4.jpg
hogen 4.jpg (101.83 KiB) Viewed 6544 times
Let's now project from the elevation view over to the board we wish to layout upon, starting with the lines which will give the face cut:
hogen 7.jpg
hogen 7.jpg (154.2 KiB) Viewed 6544 times
The dimension 'x' gives the cross-section width of the other piece to which we are connecting.

Similarly, we can project lines for the face cut miter, noting that we take these lines in reference to the board we are laying out:
hogen 8.jpg
hogen 8.jpg (168.21 KiB) Viewed 6544 times
Connect the dots and we have our face cut lines:
Hogen 11.jpg
Hogen 11.jpg (125.18 KiB) Viewed 6544 times
So, the above steps to start should be much as one would expect, having detailed a similar method for the hopper face cut layout in tAJCD volume II.

Now I'll insert a plane at slope with the face and seat cut already made, so it can be compared to the elevation view:
Hogen 10.jpg
Hogen 10.jpg (170.26 KiB) Viewed 6544 times
Let's zoom in a little closer so we can see what is happening:
hogen 9.jpg
hogen 9.jpg (183.4 KiB) Viewed 6544 times
When we take a component length from the unit triangle (drawn in an elevation view), and place that component at slope, the length will become the next larger related component in the unit triangle. They are like secants to one another. For example, if you took a dimension for chōgen (長玄) in plumb and tilted it to unit slope, it would become the same length as run of the triangle, ko (殳); if we took the dimension for the run ko (殳) and placed it at slope it would become the same length as gen (玄). These points were clearly explained in tAJCD volumes II and IV, especially when looking at leg length reckoning for a splayed post structure.

In this case we are not dealing with the longer sides of the triangle undergoing change when tilted, but the short side, or (勾). The next largest related triangle component for that measure is hogen (補玄), aka the 'supplemental' slope. Thus to lay out the height (勾) found on the elevation plane onto the actual board face (which is tilted), we must use hogen (補玄) on that tilted plane.

The key point is to be clear on what the elevation view is representing and what happens to the unit triangle parts which associate to the elevation view when we are laying out on the actual board, which will be placed at a tilt. The tilting affects dimensions in the unit triangle which are vertical (i.e, (勾)), and not those which are horizontal (i.e., ko (殳)). Horizontal dimensions are unaffected by tilts of the board to slope.

Thus, the return lip is defined by the horizontal run ko (殳) paired with the 'stretched' rise, (勾), which became hogen (補玄).

Using projection, we can also mark the intersection of the hogen slope with the board's face cut line:
Hogen 11.jpg
Hogen 11.jpg (144.19 KiB) Viewed 6534 times
Let me know if the above was at least remotely clear and somewhat understandable.
Jack_Ervin.
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Re: Hogen

Sat Jan 30, 2016 9:46 pm

Chris Hall wrote:
Hogen 11.jpg
Let me know if the above was at least remotely clear and somewhat understandable.
I opened this up earlier and read thru it when Hogen 9.jpg was the last image. I was going to reply but I got called away and and in the meantime I opened it up and my iPad and saw that there was and edit and another image. I still have the first posting up on my Mac and now have other duties to attend to. This first posting gave me a good understanding and I think that the edit will clarify this even further but I need to take another look before I comment further.
Sorry about the rambling but I just want to get my head wrapped around what I think I am looking at.
I will respond further but initially this looks to be a good graphic explaining how it applies.
Jack
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Re: Hogen

Sun Jan 31, 2016 1:37 pm

Chris,
I copied your above posting to Pages word processor so that each graphic is on a separate page. I was able to follow along with the development of the drawing. On here its difficult for me to scroll back an forth to keep continuity of the development. With the print out I was able to follow the development so now it is understandable. Now I need to take this, open TAJCD volumes II, IV, and the mailings of the hopper to refresh that study with this material. You see the old mind has too much content to recall but it knows where to access it. I find that printed pages are better suited for me to study than electronic pages although sometimes I will open multiple windows and follow along.
Any way good material for the thought and application process.
By the way your adding the last graphic gave me understanding of this because I had missed a point when I read it the first time thru.
Jack
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Chris Hall
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Re: Hogen

Sun Jan 31, 2016 6:35 pm

Jack,

thanks for the feedback. I think the major point to get is that the elevation view does not give the lengths we need, and that once the elevation plane is tilted to slope, the unit triangle rise must bump up to the next value, which is hogen. I did not draw in the projection lines for the inside face of the stick, but could do so if you wish to see them. I'm sure you could figure it out on your own sketch quickly enough though.

It's good to build a hopper or splayed post structure once a year at least to keep your head in the game.

I now find that I can do splayed post mortise and tenon layout, with overlapping half tenons, without any need to refer to notes on the drawing execution. It feels good to gain a solid understanding of this material, even though it represents only preliminary steps in the world of compound joinery layout.
Jack_Ervin.
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Re: Hogen

Sun Jan 31, 2016 6:46 pm

Chris Hall wrote: I did not draw in the projection lines for the inside face of the stick, but could do so if you wish to see them. I'm sure you could figure it out on your own sketch quickly enough though.
Chris,
No, there is no need for the inside projection lines here. Like you said, I can work on that here. You have enough on your plate already and I could, as you said, use the exercise to brush up.
Thanks again,
Jack

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