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 (61.37 KiB) Viewed 8147 times
Now, we can represent the intersection of the parts with an elevation view:
- hogen 5.jpg (173.69 KiB) Viewed 8147 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 (101.83 KiB) Viewed 8147 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 (154.2 KiB) Viewed 8147 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 (168.21 KiB) Viewed 8147 times
Connect the dots and we have our face cut lines:
- Hogen 11.jpg (125.18 KiB) Viewed 8147 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 (170.26 KiB) Viewed 8147 times
Let's zoom in a little closer so we can see what is happening:
- hogen 9.jpg (183.4 KiB) Viewed 8147 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
kō (勾). The next largest related triangle component for that measure is
hogen (補玄), aka the 'supplemental' slope. Thus to lay out the height
kō (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,
kō (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,
kō (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 (144.19 KiB) Viewed 8137 times
Let me know if the above was at least remotely clear and somewhat understandable.