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.