Judging a fair curve can be a refined skill, and some seem better at it than others. I think most of us can spot, especially if it is pointed out, a curved line of surface which is not fair. I mean ‘fair’ in the boatbuilder’s sense of the term, that is,
A “fair curve” or line is one that is as smooth as it can be as it follows the path it must take around the hull of a boat. A fair line is free of extraneous bumps or hollows, and an unfair line needs to be faired, or smoothed out.
So, in this case the curves I want to compare and analyze are not those of a boat hull, but of the fascia elements which comprise the eave-edge build up of timbers characteristic of a Japanese curved roof.
The driving element is the lowest one, riding along the top of the rafters at the edge of the eave, termed the kaya-oi. This term literally means to “bear the weight of the straw”, a reference to it’s past use on thatched roof buildings.
In this rather large and complicated build-up, the kaya-oi is the piece at the bottom at the bottom of the stack, running along the tops of the rafters:
The kaya-oi, like the members which are placed atop it, not only curves up but as it moves along towards the hip it swells in thickness too. The swelling counteracts a visual foreshortening effect, similar to railway tracks, that happens with parts the move away from an observer’s line of sight.
The shape of the curve is a matter of aesthetic preference, or so it seems, and reflects changes that Japanese carpenters made to the temple architecture imported from China and Korea. Chinese temples employ strong curves in their fascia, even to the extent of the fascia approaching a vertical orientation towards their termination point, and it is typical also that the front face of the kaya-oi in such cases also twists as it rises.
In Japanese work, a preference for much gentler roof curves and methods of curving the roof which did not require surfaces on the kaya-oi to twist. Japanese carpentry texts show a variety of methods for producing the curve in the kaya-oi and other parts of the fascia, and these methods are described as producing a variety of curve types.
Togashi mentions three types of kaya-oi curve:
- 曲線を腰反り Kyokusen wo Koshizori
- 曲線を先反り Kyokusen wo Sakizori
- 曲線を中反り Kyokusen wo Nakazori
Type 1, as I call it, is a curve that rises quickly and has less of a sag/belly to it. Type 2 is a curve that rises slowly and has a more pronounced belly. Type 3 is a midway position between the first two.
Various methods are then shown in the texts for drawing the curves, and comments might be made in those texts as to the characteristics of that curve, such as whether the curve rises quickly or slowly, flattens out or accelerates, etc. The methods for producing the curves typically involve swinging an arc with a radius which is in a particular multiple of the rise of the curve, then then subdividing this arc into portions, either radially or vertically. The intersections of the division lines to the radius produce a series of hight points, which are then projected across a series of divisions laid out along the total portion of the fascia which is to be curved.
Quite a few years ago I made a study of this issue using a 2D drafting program, and was able to at least confirm the general comments about the shape characteristics of curves produced by different methods. The curves can be drawn by each of the respective methods, then copied to the side and overlaid in the drawing so as to compare them directly. Still, I felt there was more to do in terms of analyzing the curves to see if there might be a most ideal solution among the choices, or perhaps another better solution altogether, but for the time being shelved that work and got on with other things.
With my recent plunge into Rhino however, new avenues have opening up so I decided to reinvestigate this problem. Also, I had obtained in the past couple of years a layout text on hip rafter layout, written by a fellow named Mochida Takeo, which depicted a different method of producing the kaya-oi curve. I wanted to see how that method might compare to the others with which I was already familiar.
Going to a new CAD program has made possible both the drawing of mathematically accurate curves, and their analysis. Rhino has both a native ability to produce all kinds of curves, but it has a curve analysis tool and a curve fairing tool as well that have been most helpful in making sense of things.
By the end of the study I had worked through a bunch of different arrangements using the standard methods of which I was aware:
We’ll zoom in a bit more on individual examples shortly to see what they reveal. At top left of the overview we have two parabolic curves, the last ones that were drawn, as part of a hunch I had to check out. At the left lower portion and part of the middle row are seen the use of certain slope angles to produce divisions – a method shown by Mochida. At top middle are two uses of vertical divisions, and to the right are various types of radial arc divisions used to project points, which is generally what you see Togashi using in his own drawings, so I imagine it is what he prefers.
Around the upper middle portion of the sketch is a cluster of some of the different type of curves produced, overlaid. Here’s a closer look at that:
There are 7 lines overlaid in the same spot, each subtly different than the other. But stepping back, one can see that in fact they really aren’t all that varied from one another in overall form. That’s the point of the picture – the lines from the fastest climbing to the slowest, most flattened out and most curved, are really in a fairly narrow range.
This is slightly misleading though. The cumulative effect of several tiers of similarly-produced lines will present eave build-ups which do look more significantly different from one another than the lines alone might suggest. This is because the layers which sit atop the kaya-oi end up, due to the cumulative effect of the swelling of the parts, climbing up higher and further out than the kaya-oi, which, is after all on the lowest level. Many of the methods used to produce the slightly different shaped curves in fact have a more pronounced effect the further out along the curve you go, so this will manifest in those upper layers of eave build up. In particular, the method of dividing the arc by vertical lines will produce a big surge of acceleration in the curve as you move out beyond the tip of the kaya-oi.
But, that set to one side, the various methods do not appear to produce significantly different curves, though we must also recognize, that regardless of the size of one’s computer monitor screen, we cannot see and assess the curves quite as well as we might if we had full size pieces cut to the various curves and could sight along them. Then perhaps we could get a better sense of it, but at a cost of a lot of wood and fabrication time. Otherwise we’re just left with a bunch of curves which are pretty close to one another, and maybe one is not appreciably different than the next? Is there any other way to assess how smoothly continuous a curve might be?
Rhino has a tool for analyzing tools which I had seen described in a video about working on the curves for a boat hull. Immediately coming to my mind was the problem of assessing kaya-oi curves, so now I was excited to find that I had a new avenue along which to investigate.
How this tool works is to provide you with a division of your curve line in question to any number of segments, and then each segment is correlated to a section of circle of the same radius. Whether the radius is smaller or larger is manifested by the visual of tangential projection lines – the further out from the line that the tangents project, the tighter the curve. This simple sketch gives a visual which I hope helps show how this curve graphing function works:
The line in blue is the curve being analyzed, and the red curve and projections are the analyzer. You can see on analyzer where the curve being analyzed turns the tightest, at the right side, as the graph in that location grows largest. You can also see how the curve accelerates and decelerates in slope by looking at the graph line rising and falling.
So let’s apply this analysis to various types of curves for the kaya-oi and see what can be learned from it.
The first curve I took a look at is the one Togashi favors using, and that is an arc with a radius twice as great as the amount of total rise, and then division of that arc radially:
The swung arc is divided into the number of rafter spaces, and divided into the same number of ‘pie wedges’ as there are spaces. In this method, the reference points are on the rafter centerlines, except for the last interval to the right, which is half a rafter thickness less in length. This difference in length relates to a Japanese convention in the spacing location for a rafter at the end of the run, which due to the termination of the kaya-oi run at the hip would be too short to use. So that last apparent rafter is only a place marker, but it does mean in the above case that the last spacing is inconsistent as well, crowding in at the very end.
Turning on the analyzer shows the flow of this curve:
As you can see, the curve climbs unevenly, flattening out at the second rafter centerline, then abruptly accelerating to the next, only to ease off toward the end.
I’ve often wondered if the technique used above, where the last interval is shortened by half a rafter, might lead to an uneven curve, and it seems to be the case.
So, I reconfigured the drawing, keeping the same rise but bringing the lateral intersection points back so that all of them are at the side of the rafter instead of the centerline. This is a method used by some other folks.
Here’s the redraw of the curve, shown in the upper portion of the following sketch, using the adjusted intersection points and with the analyzer turned on, and the preceding curve left on at bottom for the sake of comparison:
This change in that last space had clearly led to some improvement, as the analyzer shows. The curve climbs quite evenly, save for the second to last rafter, where it flattens out slightly.
Now, Rhino also has a tool called ‘Fair curve’ which will analyze the curve shape and make slight adjustments to fair it out. Let’s see how it changes the result:
As you can see, the faired curve (at the top) is not much different than the un-faired curve below. It has been slightly smoothed out but retains the kink.
One can zoom in on the original intersections points to see what effect the fairing has had:
If you notice the small point at the end of the horizontal line coming in from the left, you are seeing the point generated by the drawing method. As you can see, fairing the curve in this location has involved pulling the curve line down slightly from that point. In other locations one will see that the curve line has been bumped up slightly after fairing. This is much the same as one might do when playing around with the same thing on a layout board with nails in place around which to bend a batten.
The next method we’ll look at is that of using a circle divided radially as before, but in this case the circle’s radius is the same as the rise amount, instead of double as with the preceding example.
Along the left side of the picture you can see the two different size circles used to generate points, small circle at top, double-radius circle below. Notice how the curve from the small circle rises more quickly than the one from the large circle, and has intensified the kink in the curve line. And fairing that line has much the same outcome as fairing the line generated with the arc having a radius twice as large as the rise (not illustrated).
It seems like what associates to using a ‘circle divided into pie wedges’ method is that the curve produced will have a kink at the last actual jack rafter and even when faired the result remains.
Let’s look at another method then. In this one, the arc having a radius twice as large as the curve rise is employed, except that this arc is then divided with evenly spaced vertical lines instead of radial ones. I’ll show the result, with the stock version in the lower half of the picture, and the faired version in the upper half:
As you can see, this method produces quite a different curve than the radial division method. The curve climbs gradually until the third rafter over, suddenly accelerates to the last jack rafter, only to flatten out again at the end. This type of curve is what Togashi terms kyokusen wo sakizori, what I would call a type 2. The fairing tool again serves to smooth out the line but does not get rid of the unevenness altogether.
The next type of curve is produced via a method shown in Mochida’s book, and, like Togashi, he devotes a few pages in his book to comparing methods for producing kaya-oi curvature. This method involves the employment of a slope line, which is then divided into even portions, the division lines projected to an arc, and then laterally across to the relevant intersection points. In the next picture we can inspect the result, both that of the method directly (bottom of the picture) and after employing the ‘fair curve’ tool (top):
Interestingly, while the method produces a kinked curve just like Togashi’s preferred method, the kink produced by this method comes one rafter earlier. Unlike Togashi’s preferred method, this curve, when faired, becomes exceptionally clean and even. I was surprised and impressed by this result.
I also explored drawing curves using slightly different slopes to see what the effect might be. While that won’t be illustrated here, I can say that the steeper the slope use, the more the curve is flattened out, while the opposite holds true for employment of a slacker slope.
Now, one thing I have been wondering, especially after looking at all the curves superimposed upon one another and seeing how essentially similar they are to one another, is whether the various methods involved in their production might in fact be aiming at pretty much the same thing, and just not quite able to produce the perfect curve perhaps in any case. It might be the case that there is a mathematically perfect curve one could use, but that the methods that carpenters use is some sort of shorthand approximation that they can realize with conventional layout tools instead of mathematical formulae. Particularly when you look at carpentry methods used way back in time, when a much lower percentage of the population had access to schooling, and the carpentry apprenticeship began well before a young man would have even delved into slightly more complicated mathematics had they remained in school, well, you can see how it would come about that non-mathematical methods would be those employed in a carpentry shop. Even today, with mandatory schooling to grade 12, the vast majority of the population emerges with very poor mathematics knowledge.
So, one thing I’ve been familiar with for a while is the use of parabolas in Japanese traditional castle architecture, where the shape was employed in the foundations of a castle. Hiroshi Yanai’s work, Parabola Drawing Methods in Japanese Traditional Architecture, references a 17th century document from the Gotō family showing a method to draw a parabola with strings, which may trace back to method shown in a Chinese work from 1078:
Similar techniques are employed for producing the curves on Japanese gable ends, whether curved concavely (chidori) or convexly (mukuri).On a hunch then, I decided to draw the kaya-oi curve line using a parabolic curve given the constraints in place, namely total length, total rise, and number of divisions. Working this out mathematically, I produced the following curve (below) and then faired it (above): You can see immediately that the parabolic curve is perfectly smooth. Hitting the ‘fair curve’ tool made absolutely no change to the curve or the curve analyzer result. I zoomed in and could see that the faired line was on the exact same spots. That tells me that, by the software’s determination at least, the fairest curve was the parabolic one.When all the curves are overlaid upon one another, the parabolic curve sits in the happy middle ground between the various curves produced by other methods. This zoom in shows the spread of lines, with the parabolic line in black ink:
The purple line at the bottom with the most sag in it is the one produced by the vertical line division method, while the flattest line at top, in orange, is the one produced by the small circle, radially divided.I tend to conclude that the various traditional drawing methods are trying to approximate a parabolic curve, and do so fairly well but with various quirks. Now, from where I sit, knowing the above information obtained by curve analysis in Rhino, when I go to design a curved eave for a Japanese building, I am inclined to select the method where a true parabolic line is produced. There is no fairer line. But in truth, the other methods aren’t all that far off the parabola, and maybe one could make an argument against using a ‘perfect’ curve line, and instead opting for one which is not so perfectly smooth to show, perhaps, to those few who would notice, the delight that comes with imperfection.If it were the case that I wanted to avoid any use of mathematics to produce a kaya-oi curve, then my choice would be to use Mochida’s method. That method is closest to the parabola, though close inspection of the two curves overlaid reveals the parabola to have slightly more sag along the run.
There is a follow up to this post.