## Stereotomy

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

## Thursday, September 29, 2011

### devers de pas (2)

In the first post in this series, I introduced the "devers de pas" method and gave an example of its use. Now we will see how to construct the footprint at ground level in a few different situations. In future posts I will cover more complicated cases.

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
plan:

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:

Now, draw a perpendicular line to ML at M and mark the point where it intersects our baseline KL as "Q":

A perpendicular line dropped from surface in an elevation view is very common and is labeled "T.C." for "trait carré" or square line. Let's also flip that into 3D:

Now it becomes obvious why the T.C. line is so important: it's normal to the roof surface. Therefore it lies in the plane passing through G that we are searching for:

So we can draw the dévers de pas line in our drawing:

and fill in the footprint of the rafter. We give the rafter a width along the gutter line and use the inner surface intersection from the rafter elevation, as before:

In the final 3D view, we see how the plane we constructed is parallel to the sides of the rafter:

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.

## Friday, September 23, 2011

### devers de pas

I first discovered "Traité Théorique et Pratique de Charpente" by Louis Mazerolle on Chris Hall's blog the Carpentry Way in an amazing series of posts. Mazerolle was a "compagnon de devoir" or master carpenter in the 19th century in France, and his book is a compendium of complicated and obscure drawings of roofs, buildings and staircases. In this tradition the plan of a roof would be drawn out at full scale on the ground; then, the individual timbers would be placed over the plan, scribed, and then cut. 112 plates take the reader from fairly simple dormers and roofs to sawhorse challenge problems and incredibly baroque curved structures. Chris built a crazy sawhorse from the book and worked through several of the drawings using SketchUp, but ultimately put that aside in frustration. I was intrigued and got a copy of the tome (it's big and expensive, even in France). I started doing some of the same problems and shared my results with Chris, with the happy result that he's started blogging about it again. We are collaborating on deciphering this work and, based on our drawings, building 3D models of the structures to verify that we've gotten it right. Chris' frustration was justified. The text is extremely terse and filled with small typos, usually references to non-existent or wrong labels on the drawings. While the drawings are beautiful, the details cannot be trusted. Straight lines aren't, right angles aren't, some things are just plain wrong. I suspect that a lot of this was introduced in the process of recopying existing drawings to make the engravings. It would not be possible to build a lot of the structures directly from these plans; on the other hand, that is not the point. An apprentice would redo the drawing using the appropriate techniques and thus learn his art. When I started in on the book I realized that, though the techniques used to produce the drawings were not well explained and often obscure, they are basically sound and make sense to someone with a background in math or computer graphics. Furthermore, they become a lot more clear when one can whip up a quick 3D model to understand the 2D construction. I thought it would be interesting reconstruct some of Mazerolle's plates, with a more complete explanation of the geometry and some nice 3D graphics. This post is my first attempt at this, but it is a sort of prequel in that it tries to present the motivation for a construction, called "devers de pas," that is explained early on and then used throughout the book. "Devers de pas" literally means "footprint area," though "level section" might be more idiomatic in this context. "Devers" is actually an archaic French word that has just about disappeared, except in certain locutions, that means "pertaining to," belonging to," or "nearby." It's very close in spelling to another word "dévers", meaning "angle" or "angled," and we will actually be talking about "dévers de pas" in a bit! To top off the confusion, Mazerolle spells "dévers" as "devers" everywhere, which must have been a regionalism. The technique makes use of the footprint of a solid sitting on a horizontal plane to reason about its 3-dimensional geometry and the cuts needed to arrive at its final shape. In carpentry drawing, the solid is a piece of wood, usually some kind of rafter. However, the technique is very general and can be applied in many different contexts. As an example, here is the plan of a roof model that we see more of later in the book. Never mind the rafters poking through the surface of the roof. The top part of the plan shows the principal rafters i.e., the rafters on the sides. This establishes the height of the roof peak and the slopes of the side roof surfaces; it also indirectly establishes the slopes of the top and bottom roof surfaces, which are different from those of the sides. and here's a 3D view. We are more concerned with the rafters than the roof surface. Let's take a look at the footprint of the left (as seen from the top) rafter. Actually, we are more interested in the edges of the footprint than the footprint itself. Each edge is the intersection of the ground plane and the corresponding face of the piece of wood. So, the edge is a line that is in both the ground plane and the surface plane. This is still true if we extend the line of the edge past the boundary of the footprint: The extension line is called the "dévers de pas" or "angle projection" and will be annotated as "DP" on our drawings. These DP lines represent the intersection of a plane with the ground plane and, as we shall see later, can be used to draw the intersections of planes and the resulting solids. The plane of the side of a plumb rafter isn't too interesting, so let's look at the skewed rafter in the top roof surface. The DP line isn't coincident with the plan of the edges anymore. Here's the situation in 3D: In this case, the drawing of the footprint was derived from the DP line, and then the edge lines could be drawn on the plan. In the next post on the subject we will see how to construct DP lines in different situations, including this one. Sometimes the footprint is determined by other factors and determines the DP line, other times the DP line comes first. As a final teaser, we'll see how the devers de pas method can be used to make the cuts for the left face (as seen from the top) of this skewed rafter. GC is the top edge of the left face. On the plan we drop a right angle from the dévers de pas line through C, giving us the point T. We can get the height of point C in three dimensions from this drawing, but I won't do the construction now. Suffice it to say that I found the height, then drew that from C at right angles to TC, giving us U. Connect T and U to form a pointy right triangle. Also note that I've marked the intersection of the DP line with the surface of the left rafter as Z. Here's the situation if we fold up the triangle in 3d: We see that TU [correction: I had written "ZU" here] is the slant distance from T to the top of the left edge. Since that point is on the left surface, and the DP line is also in the left surface plane, The resulting right triangle is as well. We can draft that in 2D by swinging the line TU' around to be coincident with TC. If we flip that up into 3D: we see that GU' is the same length as the top edge of the rafter's surface. Furthermore, the angle U'GT is the angle with the ground on that surface, and GU'Z is the cut angle at the top of the rafter. This is even more clear if we take an orthographic view of the rafter and this plane: We've had a very brief look at the power of the "devers de pas" method. I hope this whets your appetite for more. Chris presents a clever use of devers de pas in his series of blog posts called X Marks the Spot, which presents a challenge problem of determining the intersection of two rafters at arbitrary angles.

### Welcome

Stereotomy is the art of cutting a three-dimensional object using a 2D plan. I'm a computer graphics guy, and I think these old methods are very cool. I'm interested in the drawings made by the Compagnons du Devoir in the 19th century and will be exploring their techniques from my own point of view.

Today the term is used in architecture and construction history to refer to stone cutting and structural calculations that were made with graphical methods. I think that's cool too, and plan to take a look at the work of Monduit and Philibert de l'Orme. Basically, old technical drawings are cool.

"Stereotomy" is also an album by the Alan Parsons Project, but I won't be talking so much about that.