Unwrapping the OnionIn 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:
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:
Back to FootprintsWith 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:
At this point in 3d, we can place the rafter in the model and see what we have:
To find the last vertex of the footprint, we transfer the last two edges:
In 3D, we see how the lines in the cross section come down to give us the edges of the footprint:
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:
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 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.