A Real Roof Model

Faithful reader Rob Simpson made a cardboard model of the roof used in the devers de pas series of posts:

Sometimes a physical model (and a nice glass of beer) are the best aids for understanding roof geometry!

Le Piquage, or Theory of French Scribing

We are going to start with some informal observations about the geometry of intersecting timbers and, using those, explain le piquage, or "French scribing" as it is known in the English-speaking world. This layout technique is much older than the rembarrement we looked at last time. There won't be any carpentry drawing today, but I find the technique fascinating, and we will soon tie it back to a drawing done at full-scale on the ground of the building site.

Here are the king post and a hip rafter from the model in the last entry:

The two timbers are not parallel, so we can take a long edge from each and displace them in space so that they intersect. Those intersecting lines define a plane. We can move the lines anywhere we want, so we can define an infinite family of planes that are all parallel to the edges of the timbers. Here are two such planes:

The plane on the right passes through the intersection of two roof surfaces on the hip rafter. We will use it as a reference to locate features on the two timbers. The left-hand plane is not very special; it is merely near the two timbers. But, its existence implies that any two timbers can be assembled against a plane. The planes and timbers can be tilted any way we like: for example, onto the ground, over a full-scale drawing of the layout:

(I've eliminated the carved top of the king post, which can be done later). This is a very useful fact for the carpenter. It means that any two timbers can be trial fitted on the ground, using a template, without needing to erect them into their final orientation in 3D space. The picture also presents the tantalizing prospect of being able to directly lay out the joint between the hip rafter and the king post, if only we could get them together into their final arrangement.

We can't do that before the joint is cut, because the timbers are solid. The closest we can do is move one of the timbers vertically, keeping the same orientation and longitudinal rotation. In other words, working backwards in time, we displace the hip rafter upward and fill in the joint:

On each timber we've drawn a line where our reference plane intersects them when they are assembled. If we choose a point on the hip rafter's reference line and move straight down to the level of the king post reference line, we will find a point on (or near) the king post that is coincident with the upper point when the timbers are joined together in their final arrangement. Also, we make a critical observation: each pairwise combination of faces in the hip rafter and king post have the same relative orientation in this configuration as they do when together, assuming everything is straight and flat. This implies that the intersection line between two displaced faces will have the same direction in space as they do in the final assembly, though the line will of course be in a different position. More formally, the direction of the intersection line is the vector cross product of the surface normals of the planes, and the normals are direction vectors without a fixed origin, so they are unaffected by the translation.

Let's see how we might lay out the joint, starting with the side of the hip rafter that is plumb i.e., the bottom in the final assembly. I am going to assume that the operations I describe can be performed accurately by a carpenter and save discussion of possible errors until the end. After the timbers are established in their correct relative orientation, the first step is to create a vertical reference line:

This is done with a special carpenter's plumb bob with a hollow center so that it can be aligned with lines on the ground. Looking down from above, the plumb line is aligned with the intersection of the hip rafter face and protruding arris on the king post. The vertical plane defined by the plumb line and rafter face is projected and drawn on the king post with the use of dividers.

The horizontal distance from the plumb line projection on the king post to the plumb line is measured:

That distance is transferred and marked on the hip rafter.

That point marks the location where the king post arris intersects the rafter's face in the final layout.

Next, we find the direction of the intersection line of the two faces by laying a straight edge against both faces:

We draw a line on the hip rafter, but it is obviously not in the correct position. We know that the true intersection line must go through the king post arris, so we draw a line parallel to the direction line that goes through the point we marked previously:

We now have one side of the joint marked out.

The intersection line on the king post is coincident with the perpendicular line in this case, because the rafter surface is plumb. In order to find where the second side of the joint intersects the first, we measure the distance from the king post reference line up to the arris along the intersection line:

And mark it on the rafter:

That point is also where the intersection line crosses the plumb line reference. That is no great surprise because the plumb line touches the corresponding point on the king post, but this will not be true in general.

Now we establish the intersection line direction for the upper face of the king post:

And move it to cross the line from the lower plane at the correct point:

To summarize, I've redrawn the complete layout on this side without the construction lines:

Now we turn our attention to the other side of the hip rafter. Here the situation is slightly more complicated because there are two faces to lay out on the hip, and they aren't plumb. Nevertheless, the principles are the same. Start by establishing the plumb line reference against the arrises of the two rafters:

Drawing the projection of the plumb reference plane on the king post will be a bit more artisanal because there is no vertical plane to use as a reference.

The common reference point is measured and marked:

The direction of the intersection line between the two lower faces is found:

and transferred to the known intersection point:

Next we find the intersection line between the lower king post and upper rafter faces:

At the same time we find the distance along this line between the horizontal reference plane and the upper king post face. On the other side of the joint we were able to measure distance along the actual intersection line on the king post; here we cannot. However, we are measuring the distance on a line parallel to the real line, so the distance will be the same. Assuming everything the faces are flat, of course.

The intersection line is moved to its correct position on the upper hip rafter face, and the distance we just measured is marked:

The final intersection, between the two upper faces, is found:

And transferred:

Voilà the joint layout on this side:

To finish up we connect the layouts from the two sides across the top face:

and the bottom:

The connecting lines are parallel to the king post edges, which confirms that we did the layout correctly.

This is the method that built the cathedral roofs of England and France in the Middle Ages. It is much more "concrete" than the rembarrement method we looked at last time: instead of dealing with abstract planes and using descriptive geometry to find their intersections, the actual timbers are directly used as a sort of graphical 3D calculator. The method relies on principles of geometry, but those do not have to be understood in order to apply it. On the other hand, quite a bit of skill would be required to carry out a real layout, and acquiring that skill was obviously a big part of the carpenter's apprenticeship.

This post is called "Theory of French Scribing" for a good reason. We have skipped many details of the process in order to concentrate on how the simple application of geometry leads to a layout method. The second case we encountered -- two timbers with no plumb faces -- is actually the most difficult to handle. It would be much more common to work with timbers with plumb faces, such as when laying out a principal rafter against the king post, or other parts of the frames. Here are some of the issues we blew off:

• We assumed that we can mark and measure horizontal and vertical lines in space by eye. This obviously takes a bit of practice. Today you can buy dividers with built in spirit levels, intended for scribing the joints in a log cabin. Medieval carpenters didn't have those. On the other hand, small errors doesn't necessarily have much of an effect on our measurements. When we measured horizontally from the king post reference line out to the plumb line, we would have to be 8 degrees out of horizontal in order to have a 1 percent error. If an apprentice's dividers were that askew, the master would surely throw something at him!
  
• We assumed that the timbers were straight and their faces flat. As I said the last time, one of the strengths of scribing is that it can handle warped timbers. An out-of-square timber is a special case of the general problem of laying out oblique faces that we just did, but in practice carpenters would use a faster method to deal with that. The intersection lines we found with the straight edge are usually close to the true intersections, so the results will be close, but judgment and skill are required.
  
• Any errors we did make could be expensive. In practice, some margin would be left and trimmed away in the final trial fit of the two timbers.
  
• Finally,this is personally theoretical because I've never done it! The pages of Mazerolle that explain the method are quite confusing. There is very little free information available on the Web on French scribing, and practically none explaining the method for oblique faces. I could be making a number of false assumptions.
  

Nevertheless, we now know enough about scribing to be able to understand the notation on carpentry drawings that are intended for that layout method. We said nothing about how the timbers are positioned before scribing. We will cover that next time and see how to go from the drawing -- done at full scale -- to the arrangement and relative rotations of the timbers. I teased you by displaying a plan under the timbers, so that should give you a hint about positioning the timbers in their common reference plane.