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

Monday, October 17, 2011

devers de pas (4)

In the last post we looked at a "folding" method of developing the ground plane footprints -- devers de pas -- for a timber that works with the actual cross section of that timber. Now we will apply that to several complex shapes for the last rafters in our little model. First though, a correction...

Unwrapping the Onion

In the comments to the part 2, Chris Hall pointed out a problem with the inner surfaces of the near and far roof slabs: they do not intersect the left and right slabs at the ridge lines (in plan view). I gave the rafters at H and G the same width as the principal rafters at E and F. This does not affect our exercises for finding the footprints for these rafters, but it messes up another aspect of the layout. Consider the solid made by the inner surfaces of the rafters. If this is congruent to the shape of the main roof, then not only do many aspects of the layout become much neater, but ideals of symmetry (and, I suspect, various spiritual ideals as well) are satisfied. In a real roof there are several such solids formed by the inner and outer surfaces of the common rafters, purlins, and principal rafters, and they all should be congruent. The desired configuration in our model looks like this:
Notice the similar shapes of the inner and outer surfaces, with all the rafters lying between them. In solid geometry terms, the inner surface is the same as the outer, scaled uniformly about a point in the ground plane lying directly below the peak of the roof. The inner solid produced by my original layout, which I'm not illustrating here, would be an oddly skewed version of the main roof surface.

How do we layout the correct inner surface of the plan? We said that the inner surface has been scaled around a point underneath the peak, so the inner peak must also be directly under the outer peak. Therefore, in each elevation view that cuts through the two peaks, the distance between the two will be the same. The dimensions and layout of the principal rafters are given, so the widths of the rafters in the others surfaces will be determined by this principle. Here's a corrected version of the plan:

An interesting result is that the distance between the inner and outer surfaces will be different for each different slope in the roof, and so the rafters will all have different dimensions. This sounds like a lot of work for the carpenters, but I suppose it doesn't matter if you have to resaw everything anyway.

Back to Footprints

With that out of the way, we return to laying out rafters. The next rafter will have a completely irregular cross section with one side lying against the near surface of the roof. We start as we did for the rafter at D by drawing an elevation view of the rafter and folding a perpendicular cross section plane down to the ground:
and then find,in the cross section, the edge that lies against the near surface. This is the same construction we did last time, running a line from point 2 to the intersection of the folding line and the gutter line:
Next we draw the rest of the cross section:

At this point in 3d, we can place the rafter in the model and see what we have:
Next, we find where the points 4 and 6 end up in the footprint:
Point 9 is found by extending a parallel to the elevation cross section -- the ridge line -- through point 6 to the edge of the roof. To find point 11, we do the same construction to transfer a surface intersection from the cross section view to the plan, only in reverse. The line between 2 and 4 is extended to intersect the fold line at point 8; the line from point A to 8 is then the left edge (and dévers de pas line) of the footprint. Point 11 is located on it with the same parallel method used to find point 9.

To find the last vertex of the footprint, we transfer the last two edges:

The intersection of the edge between points 4 and 5 is outside the crop line of the plan, but I assure you that the construction is still correct :). The edges of the footprint extended from points 11 and 9 intersect to give us point 12.

In 3D, we see how the lines in the cross section come down to give us the edges of the footprint:

In passing, Mazerolle describes a bit of geometrical trickery that would allow us to find point 2 and the initial edge without drawing an elevation of the rafter. We drew the elevation view of the near roof surface when we constructed the rafter at point H. We will use it again to draw an arc centered at K and tangent to the outer roof surface:
Now, the plane of the elevation view at H passes through point K (on the ground) and is perpendicular to the roof surface. The cross section plane of rafter A also contains K. It is perpendicular to the roof surface because by definition it is perpendicular to the upper edge of the rafter, which lies in that roof surface. So, the rafter cross section plane must intersect the roof elevation at the normal line from K to the roof surface:
Therefore, the distance from K to the roof plane will be the same in the two planes. We run a line from the intersection of the fold line and roof gutter tangent to the arc we drew back to the ridge line, which gives use point 2 and the first edge.

I don't see that this is easier than drawing the elevation, but the geometry that supports this construction is interesting.

The next rafter will have an equilateral triangle as a cross section. Mazerolle says that "The rafter B ... is obtained in the same manner as that which came before." That's not exactly true in the case of the development in the book, and we'll also add an additional twist that forces us to proceed differently: the bottom of the rafter will be level. We do start by constructing an elevation view of the rafter at B:

If we ran the fold line for our cross section view through the center point K, that would put the view rather far from point B, where we will draw the footprint. In fact, we don't need to draw the fold line through K; any perpendicular to the ridge line will do. So, we've chosen a fold line that is closer to B and get our initial point.

We will give the triangle a size such that the footprint of the rafter will just touch the intersection of the inner surfaces of the left and far roofs. This implies that the rafter will "run up" the edge formed by those surfaces; therefore, in the cross section view, the lower edge of the triangle will also touch that intersection. We know how to find those surfaces and intersections in the cross section view:

We find the intersection of the fold line and the outer roof surfaces and connect them to the starting point. Then, the inner surface lines in the cross section come from the intersection of the fold line and inner ground lines which must be parallel to the outer surface lines. Their crossing at the ridge line gives us the position of the lower edge in the cross section. The cross section looks good in 3D:
The footprint is established with the same methods used before:
The two outer edges pass through the intersections of those edges in the cross section and the fold line. The end points of the third edge are found with parallels through the cross section view. As expected, the footprint just touches the intersection of the inner ground lines. The footprint agrees with the 3D view of the rafter:
The last rafter, erected at C, is a hexagon that lies against the far roof surface. The cross section and footprint are constructed using the same methods we've used already:
As we've come to expect, the 3D view shows the rafter sitting nicely on its footprint:
As a justification for fooling with such an exotic footprint, Mazerolle makes the point that if the rafter cross section was circular, then the inscribed hexagon could be used to find the elliptical footprint, with the help of a pistolet or French curve:
The draftsman would run the French curve through points of the footprint to find the shape that is known to be an ellipse. I don't have a set of French curves handy, and they would be awkward to use with Blender, so I made the ellipse by rotating and scaling a circle. That was tricky, because none of the lines between vertices of the hexagon lie on the major or minor axes of the ellipse. We could probably change the orientation of the hexagon a bit to make this method more practical.

We are now equipped to find the devers de pas footprints, and the corresponding dévers de pas surface lines, in just about any situation. While Mazerolle's carpentry drawings usually use the first technique from part 2, where we used a trait carré normal line from an existing elevation view, we should now be able to handle whatever he throws at us in the world of devers de pas.

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.