Our plan is the same as we used in the first post.
This shows a trapezoidal ground plan and a cross section of the roof at the principal rafters, which are placed points E and F. The result will be a pyramid with the faces meeting at the highest point, K. We draw in the ridge lines where each face meets its neighbor:
and take a look at the situation in 3D:
The left and right (looking from the top) roof faces have the same slope, but the top and bottom faces are at different slopes.
We are going to run rafters along all the faces and all the ridge lines, with different cross sections too. We will assume that each rafter goes all the way to the peak of the roof, even though that would produce an impossible traffic jam. Once we've found a timber's footprint, we will show it in 3D with the top sawed off. We will first find the footprint of the principal rafters:
This should be pretty easy. The top and bottom edges of the footprints are already shown on the plan: those are the vertical faces of the rafters. In order to find the inner edge of the footprint, we need to look at the elevation drawing. We find the point where the inner edge of the rafter intersects the ground plane, and run it down into the
and we get the footprint in fuchsia. Next, we will erect a rafter from point H up to the peak, with the same width and depth as the principal rafters:
This rafter has plumb faces too and won't be much harder. The only problem is that we don't have an elevation view of this roof surface. So, we construct one using the height h of the roof peak:
We constructed the elevation right over the plan. This is common in the French carpentry drawing tradition. It saves space, which is important if the plan is drawn at full scale (called an "épure".) It also allows us to easily move between the plan and the elevation. Again, the inner surface intersection line gives us the inner edge of the footprint.
When we set out to find the "dévers de pas" line that defines the side of a footprint, we are really looking to define the plane for that side of the timber. In order to draw the line on the plan, we need to connect two points that lie on both the timber's side and the ground plane. We usually have one point in hand already: The point where the side meets the roof gutter line. So, the problem often reduces to finding one more line in the side plane and its intersection with the ground.
Consider the principal rafter rising from point E. The rafter's sides are square, so if we drop a perpendicular line from a point on the top edge, it will lie in the side face:
Since the upper surface of the rafter lies in the surface of the roof, that line is a surface normal, and is parallel to all other surface normals of that roof:
This is good stuff. If we need a line for our dévers de pas construction, we just need to look for a surface normal somewhere.
Next, erect a rafter from G up to the peak. This is our skewed rafter from the first post:
We are not going to construct the projection line directly on the side of the footprint yet. Instead we will construct a parallel plane -- and D.P. line -- starting from G because that works better with our principal reference point, the peak of the roof at K. That plane contains the line from G to K; we are looking for one more line to completely specify it.
As before, we start with an elevation view of this roof surface. We can use the roof height, but the view needs to be along the roof line. We do that by running a line perpendicular to the roof edge through K and forming a right triangle with the roof peak height, taking us to point M:
If you tilt your head over to the side, you are looking at the elevation view of rafters in this roof surface. Even if a rafter doesn't rise square from the gutter line, it will usually be kept between the upper and lower surfaces shown in this view. If we cut out the elevation view and tip it up in 3D, we can see what's going on:
This construction that uses a surface normal in the form of the "trait carré" to derive the D.P. line is used again and again in Mazerolle's drawings.
P.S. Chris Hall has a go at the same model in this post. Check it out; it was one of my inspirations for learning more about this French method and Louis Mazerolle's works.