devers de pas (3)

In this post we will add only one rafter to our evolving model, but we will show two different ways to determine its footprint -- or devers de pas -- on the plan. We are going to add a hip rafter at the intersection of the right and near roofs. We will keep this rafter square, so we must choose which roof to align it with, if any; we will orient it so that its top face aligns with the right roof surface.

In the plan, we show the roof ridge lines. One arris of the rafter will run along the ridge line from D at ground level to K at the roof peak. We will set the width along the gutter line of the rafter, rather than sizing the rafter itself. In my model I've set that width at 1.5 cm and marked the end of the edge along the gutter as I.

We proceed exactly as we did with rafter G in the last post. In fact, we can do even less work, because we don't need to construct an elevation view of this roof surface: we already have the elevation view at the top of the plan. We will use it to construct the surface normal by dropping a perpendicular -- trait carré -- from the top of the roof line in the elevation view. We draw the intersection at point R.

In this roof construction the elevation view is only valid for a cross section taken through the point K. For simple A-frame style roofs and many more complex shapes this principal elevation view might be useful all along the roof, but the difficulty is the same: we can't use R directly, because the elevation view is placed arbitrarily high in the drawing. It is aligned with the center axis of the plan, so can drop a vertical line from R to the point R', which lies on the line where the elevation view was made, to find where the roof surface normal intersects the plan.

A 3D view:

We know that we can connect D with R' to get the dévers de pas line at the intersection of the near surface plane of the rafter with the plan:

and the 3D view of that plane shows that we did this correctly.

We finish the footprint on the plan by drawing a line parallel to R'D through point I. The inner (left) face of the rafter intersects the ground on the line determined in the elevation view for all rafters in this roof.

We will now draw the rafter footprint by another method which, while it requires more work, is also more versatile. We will construct the cross section of the rafter, project those edges onto the plan, and then connect them up to find the footprint. The first step is to construct an elevation view of the rafter:

Using the height h of the roof, we extend a perpendicular line from KD to point S. DS is a view of the arris of the rafter that runs along the ridge line. Next, draw a perpendicular line to DS running through the center point K, which intersects DS at T. Drop an arc KT down to KD, which represents the ground in this view. The arc intersects KD at point U.

We are laying out a view of the cross section of the rafter. It will be most convenient to draw the cross section on top of the rest of the plan, even though will need to be careful to keep these two views straight. In 3D we can see that we are taking the cross-section plane of the rafter that passes through the center point of the ground plan:

and folding it along the line KS down to the plan:

U was on the top edge of the rafter, and now we are working with it on top of the plan. How does the view of the cross section plane and the rest of the plan relate? We obviously can't carry points directly from one view to the other, but we can make two helpful observations. First, any point on the line KS will be the same in both views, because that was the axis of rotation of our fold. Second, distances measured perpendicular to KU will be the same in both views because KU is perpendicular to the folding rotation. This is equivalent to stating that parallels to KU are the same in both views.

The next task is constructing the upper edge of the rafter on the cross section. This is the major constraint in the layout of the cross section because the rafter is specified to lie against the right roof surface. This edge is the intersection of the cross section plane and the roof surface; in order to draw it we need to find two points on that intersection and connect them. The first point is U. The second is the intersection of the fold line KS with the gutter line of the roof surface. They intersect at V:

in 3D:

The lower rafter surface also intersects the cross section plane at a point on KS, which we mark as V'. The intersection line most be parallel to line for the upper surface, KS, so we draw that passing though V':

The cross section is rectangular, so make a perpendicular line to UV at U. This line intersects the inner surface intersection line at X, giving us another point on the cross section:

It intersects KS at y, which is of course in the plane of the ground plan:

Now, using the fact that parallels to KU are the same in both views, we extend a line from I to intersect UV at point a:

This gives us a 3rd point on the cross section. Drawing a parallel to UX through a gives us point z and the complete intersection.

As one would hope, this cross section aligns perfectly with the actual cross section of the rafter:

Back to devers de pas. Point y lies in the plane of the near surface of the rafter and in the ground plan, as does D. So, we connect them to get the dévers de pas line Dy at the intersection of the near rafter plane and the ground:

In fact, Dy does run along the near edge of the rafter footprint:

We finish the footprint on the plan by using a parallel to Dy running through point I, and it agrees perfectly with the footprint we constructed by dropping a trait carré from the rafter elevation.

This folding method works in situations where we can't conveniently find a surface normal for a face of a rafter. Chris Hall used a variation of the technique in his X Marks the Spot series of posts. It was useful there because the orientation of the timbers was arbitrary and not related to any roof surface. It also works in situations where the rafter cross section is not square, or even polygonal (!), as we shall see in the next post. Stay tuned.