Stereotomy

The art of representing objects in section, elevation and plan in order to cut them out. - Louis Mazerolle

Wednesday, October 5, 2011

devers de pas (3)

In this post we will add only one rafter to our evolving model, but we will show two different ways to determine its footprint -- or devers de pas -- on the plan. We are going to add a hip rafter at the intersection of the right and near roofs. We will keep this rafter square, so we must choose which roof to align it with, if any; we will orient it so that its top face aligns with the right roof surface.
In the plan, we show the roof ridge lines. One arris of the rafter will run along the ridge line from D at ground level to K at the roof peak. We will set the width along the gutter line of the rafter, rather than sizing the rafter itself. In my model I've set that width at 1.5 cm and marked the end of the edge along the gutter as I.
We proceed exactly as we did with rafter G in the last post. In fact, we can do even less work, because we don't need to construct an elevation view of this roof surface: we already have the elevation view at the top of the plan. We will use it to construct the surface normal by dropping a perpendicular -- trait carré -- from the top of the roof line in the elevation view. We draw the intersection at point R.
In this roof construction the elevation view is only valid for a cross section taken through the point K. For simple A-frame style roofs and many more complex shapes this principal elevation view might be useful all along the roof, but the difficulty is the same: we can't use R directly, because the elevation view is placed arbitrarily high in the drawing. It is aligned with the center axis of the plan, so can drop a vertical line from R to the point R', which lies on the line where the elevation view was made, to find where the roof surface normal intersects the plan.

A 3D view:

We know that we can connect D with R' to get the dévers de pas line at the intersection of the near surface plane of the rafter with the plan:
and the 3D view of that plane shows that we did this correctly.
We finish the footprint on the plan by drawing a line parallel to R'D through point I. The inner (left) face of the rafter intersects the ground on the line determined in the elevation view for all rafters in this roof.
We will now draw the rafter footprint by another method which, while it requires more work, is also more versatile. We will construct the cross section of the rafter, project those edges onto the plan, and then connect them up to find the footprint. The first step is to construct an elevation view of the rafter:
Using the height h of the roof, we extend a perpendicular line from KD to point S. DS is a view of the arris of the rafter that runs along the ridge line. Next, draw a perpendicular line to DS running through the center point K, which intersects DS at T. Drop an arc KT down to KD, which represents the ground in this view. The arc intersects KD at point U.

We are laying out a view of the cross section of the rafter. It will be most convenient to draw the cross section on top of the rest of the plan, even though will need to be careful to keep these two views straight. In 3D we can see that we are taking the cross-section plane of the rafter that passes through the center point of the ground plan:

and folding it along the line KS down to the plan:
U was on the top edge of the rafter, and now we are working with it on top of the plan. How does the view of the cross section plane and the rest of the plan relate? We obviously can't carry points directly from one view to the other, but we can make two helpful observations. First, any point on the line KS will be the same in both views, because that was the axis of rotation of our fold. Second, distances measured perpendicular to KU will be the same in both views because KU is perpendicular to the folding rotation. This is equivalent to stating that parallels to KU are the same in both views.

The next task is constructing the upper edge of the rafter on the cross section. This is the major constraint in the layout of the cross section because the rafter is specified to lie against the right roof surface. This edge is the intersection of the cross section plane and the roof surface; in order to draw it we need to find two points on that intersection and connect them. The first point is U. The second is the intersection of the fold line KS with the gutter line of the roof surface. They intersect at V:

in 3D:
The lower rafter surface also intersects the cross section plane at a point on KS, which we mark as V'. The intersection line most be parallel to line for the upper surface, KS, so we draw that passing though V':
The cross section is rectangular, so make a perpendicular line to UV at U. This line intersects the inner surface intersection line at X, giving us another point on the cross section:
It intersects KS at y, which is of course in the plane of the ground plan:
Now, using the fact that parallels to KU are the same in both views, we extend a line from I to intersect UV at point a:
This gives us a 3rd point on the cross section. Drawing a parallel to UX through a gives us point z and the complete intersection.
As one would hope, this cross section aligns perfectly with the actual cross section of the rafter:
Back to devers de pas. Point y lies in the plane of the near surface of the rafter and in the ground plan, as does D. So, we connect them to get the dévers de pas line Dy at the intersection of the near rafter plane and the ground:
In fact, Dy does run along the near edge of the rafter footprint:
We finish the footprint on the plan by using a parallel to Dy running through point I, and it agrees perfectly with the footprint we constructed by dropping a trait carré from the rafter elevation.

This folding method works in situations where we can't conveniently find a surface normal for a face of a rafter. Chris Hall used a variation of the technique in his X Marks the Spot series of posts. It was useful there because the orientation of the timbers was arbitrary and not related to any roof surface. It also works in situations where the rafter cross section is not square, or even polygonal (!), as we shall see in the next post. Stay tuned.

15 comments:

  1. Tim,

    clearly explained and very well done, though I think the depiction of the rotation of the stick's cross section down to the ground could be made a bit clearer, as illustrations go.

    Looking forward to the next post!

    ~C

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  2. Chris,
    Yes, the transparency is a bit out of control and confusing. There will be other opportunities to demonstrate folding / rabattement in the future; I better go back and look at "X Marks the Spot" some more for some ideas. I remember that you demonstrated it quite clearly there.

    Tim

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  3. Yeah, I had to play around with it a bit myself to get the look I wanted. I made a series of copies by rotation and made each copy a different percentage of opacity. I'm not sure though how your drawing software works. I'm sure you'll come up with something that works.

    ~C

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  4. Tim,
    A hip rafter connects two roof planes. In this development you have used the roof plane to the right of point D, probably because this was the elevation already in the drawing. I have attempted the development using the roof plane to the left of point D, drawing the appropriate elevation and my TC meets line HG very near to point G.
    Have I made a false assumption or do you have an explanation why the hip rafter appears to have two DP's (which are not parallel) !?
    Very best regards.

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  5. Rob,
    This hip rafter has square angles, so it can't sit against both faces. That might mean that it's not much of a hip rafter, but if it did connect the two roof planes it would need backing cuts and assume a pentagonal shape. I made the rafter lie against the DF edge; you appear to be using edge DH. You are constructing the DP line for a different configuration of the rafter, so your DP line won't be parallel to mine.

    Tim

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  6. Tim,

    a couple of observations. Unlike the book, you have made your rafter footprint at point H to suit the inner track line of the commons, descending from point d, a line to have offset equally, it would appear, all the way around. While not being quite sure how you produced that line as a parallel to AD in plan, in the book, that line descending from d meets the plan line for the lower hips (lines AK and DK) and at that point turns across to parallel line AD. This results in a more truncated footprint for the rafter meeting the floor at H. Looking at your drawing you have not done this, so I'm wondering why that might be. A minor matter to be sure.

    Now, my second comment involves point I.

    As far as obtaining point I is concerned, you proceeded by setting I at an arbitrary distance along the gutter track line DC, 1.5 cm in your example. Then, after you had established your D.P. line from R' to D, you simply make a line parallel to R'D and meeting point I. All fine.

    Then you used the second method and in producing the section view UXza you simply projected from I over to line UV at point a, and so forth. Also fine.

    Here's the thing - what if a person wants to make the hip section a perfect square? Then, how would one locate point I? It seems to me that the first method, as is, will not obtain point I, and that the second method can obtain it, but only in a reverse manner to what was shown. That reverse manner is: obtain length UX, and swing that length up to line UV, and also make the same measure along line XV', to draw the perfect square section UXza. Then from there one projects back to the plan line DC, parallel to the D.P. line R'D, to obtain point I. Any thoughts on this, or see a way that the first method could produce point I for a perfectly square section?

    Mazerolle also shows a perspective view of hip DK, which could be used to locate point I, however he doesn't indicate how he located the vanishing point for that perspective view, which might be helpful.

    ~C

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  7. Chris,
    I produced the inner track line for the near edge of the roof from the elevation view that extends up from H. I know that the plate in the book uses a shallower rafter at H, but I saw no good reason to do that; I think it is more instructive to use the same sized rafter as at E and F and see how the footprint changes when it is erected at a steeper angle.

    I think the second (folding method) is probably the most natural for finding the footprint when starting from a given cross section. I think I see how to find point I using the DP line DR': find the slant line for that face of the rafter using the technique I showed in the first post and measure the desired distance perpendicular to the slant line. After some hand waving :), transfer the point thus found down to the ground and run it back to the roof edge.

    Mazerolle's "perspective" drawings are usually in cavalier projection. I don't know if you plowed through the chapter on perspective, but everything that isn't a straight orthogonal view is called "perspective."

    Thanks for the comments,
    Tim

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  8. Tim,

    you wrote,

    "I know that the plate in the book uses a shallower rafter at H, but I saw no good reason to do that; "

    Hmm, well, I think he does have a good reason to do that - use a shallower rafter at H: it is something which relates to many other drawings in the book where not only top rafter surfaces but undersurfaces are aligned to one another in planes. It's as if in drawing the roof he produces two prisms, one for the outside surface of all the rafters, braces, etc., and one for the undersurfaces of all the rafters, braces, etc., like a pair of those Russian nested dolls. note that the 'hip' rafter DK, for instance is drawn so that its undersurface meets the groundline along JN, which is the inside groundline defined by the common gf. You could have made that hip any depth you wanted, but he shows to constrain it in depth by having its lower surface meet the plane of the underside of the common.

    Thus the rafter PH ends up having a slimmer section, as it is conforming to that inner prismatic shape as are the other commons. Only with the hexagon section hip and the irregular hip does he not conform to the common rafter undersurface plane - though the hexagon does have one arris meet that inside reference line. The triangular hip BK meets the floor so that the middle of its undersurface meets the inner groundline intersection. That's deliberate.

    Working with a plane relating to the undersurfaces of the common rafters, especially in terms of how those planes meet one another at the hips, is a part of working with purlins, for instance. If you look at planche 33~34 in the book you will see a roof where all the undersurfaces of the roof parts, commons, hips, braces, etc, are all in the same plane. I'm sure there are others but I haven't investigated all the drawing yet.

    That's my take anyhow. Your reasons do make sense in terms of keeping the rafter sections constant to compare footprints, I just think Mazerolle did actually have a good reason for showing the dever de pas example the way he did. Rafter PH shows how the footprint and rafter section are interrelated quite well, when the footprint is confined to a particular place and the roof is effectively steeper.

    And to Rob: if you are still having puzzlements about laying out an alternate hip for DK, one that has its upper face aligned to the ground line AD instead of DC, then shoot me an email and I'll send you some jpegs of the DK 'hip' drawn in that manner.

    Hope you don't mind me doing that Tim! Please let me know if so.

    For Now,

    C

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  9. Chris,
    "Hmm, well, I think he does have a good reason to do that - use a shallower rafter at H: it is something which relates to many other drawings in the book where not only top rafter surfaces but undersurfaces are aligned to one another in planes."

    I would say (without checking) that all the planar roofs in the book, and the curved ones too, are constructed that way. The fact that the under surfaces of commons, top and bottom surfaces of purlins, etc. are aligned is crucial for solving the construction in the examples we've worked up together. However, that is all within single planes of each roof, and I hadn't noticed that there were constraints on the inner planes of adjoining roofs. I would have thought that the natural constraint would be to give all the roof parts the same thickness, which is what I did with the rafter at H here. But I think you are saying that the neighbor roof planes all meet at the diagonal i.e., the hip line in plan view. Wow, that implies that the timber (e.g., common rafters) will have a different dimension in each roof surface.

    Tim

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  10. "Wow, that implies that the timber (e.g., common rafters) will have a different dimension in each roof surface."

    Exactamundo! It's insane isn't it?

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  11. Tim,
    I have studied part 3 a lot, redrawn, made models etc. but I do not see why the second point on the intersection "of the cross section plane and the roof surface" is "the intersection of the fold line KS with the gutter line of the roof surface". I don't doubt it is correct, just my brain cells refuse to cooperate. Do you have a sentence or two that could unblock it for me ?

    Are you planning to tweek the series to make it a sort of tutorial introduction to Mazerolle?
    If so I have a few minor remarks* - and what about reproducing the original(s) from Mazerolle as a comparison ?

    *e.g. in figures 9 and 10 (s1600/0017.png and /0018.png I think) it would be nice if you could also draw the outside gutter line extended, then you can clearly see that intersection with KS extended.

    This series is much appreciated.
    Very best regards.

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  12. Rob,
    In order to construct a line that is the intersection of two planes, we find two points that lie on both planes. The fold line KS is in the cross section plane by definition; the point at which KS crosses the gutter line must lie in both the cross section plane and the roof plane. Does that help?

    I don't know exactly where I'm going with this. So far it has been an exercise in writing and technical illustration, and a few things are in the pipeline there.

    I haven't copied any plates yet, apart from the background of the blog, in part because I don't have an A3 scanner (coming soon, perhaps!). Also, I've been unsure about the legality of copying images out of the Editions Vial reprint, even though the images themselves have surely passed into the public domain.

    Glad you like it,
    Tim

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  13. Tim,
    Sorry to be rather late in replying.
    The "solution", your first sentence, I should have seen as it is Desc. Geom. 101. Thanks for pointing that out.
    Re reproductions, I had wondered if the copyright situation might be vague.
    On to part 4.
    Very best regards,
    Rob

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  14. Tim,

    Your step by step drawings make the Traite De Charpente drawing eaiser to understand. Here are a couple of observations that I was able to decipher from your step by step drawings.

    Drop Perpendicular
    http://www.sbebuilders.com/Traite_De_Charpente/Tim-Moore/Drop-Perpendicular.jpg

    The point U locates the dihedral angle between the plans.

    http://www.sbebuilders.com/Traite_De_Charpente/Tim-Moore/Dihedral-Angle.jpg

    Sim

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  15. Hi Sim,
    Your drawing is correct. The cross section view is constructed in the plane perpendicular to the faces that lie in the roof, so part of that includes the dihedral angle. I talk specifically about hip rafters in http://stereotomy-blog.blogspot.fr/2012/01/carpentry-model.html and use the same construction, though I don't think I identify the backing angle by name. Do you still call it a backing angle when the hip is twisted, like in this case?

    Tim

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