X Marks the Spot (VI)

Back at this gem of a French charpente drawing problem. This exercise was sent to me by a reader who in turn received it from his teacher at the American College of the Building Arts, Bruno Sutter. This is the 6th post in the series, with previous entries archived to the right side of the page.

Where we last left off, the work of marking out the basic configuration of the crossing sticks had been established in a 2D format:

Now the more challenging part of the drawing begins, which is the process of using the information we already have on our 2D to determine how the pieces intersect one another. At the end of this process we will have a drawing which we can directly measure and take angles off of, or, from which we could, if the drawing were done full scale, superimpose our pieces of wood upon the drawing and transfer the marks directly to the wood from the drawing to give our cut lines.

Let’s just refresh the memory in terms of the basic layout issue here, with the 3D sticks -note that on one stick, piece ‘a’, I have colored the face we will work on today a different color than the rest, in the interest of clarity:

In the next drawing, I have taken that orange-colored face and extended it into a much bigger plane:

While the plane can run into infinity to the left and right, it is bounded at the bottom by the floor, where the stick terminates, and by the line of cut at the top of the stick. The line along the floor coincided of course with one side of the footprint of the stick, and the line at the top coincides with one edge of the top cut of the stick. Those lines, along the floor and along the top cut, are also parallel to one another and in complete alignment.

Next I have, presto!, disappeared the orange plane and left only the lines which defined it. We are most interested initially in the plane’s trace line which runs along the floor:

Obviously, looking at the place where the sticks cross, you can see the lines of intersection formed between the orange face and two of the faces on piece ‘b’. The line along the floor from the footprint of piece ‘a’ also crosses the plan view of piece ‘b’.

Let’s take a closer look at how that line, the lower trace from the orange plane, and observe how it intersects the lines defining the plan of stick ‘b’:

Remember that you can click on these pictures to make them larger- in fact, to have any hope of getting a handle on the drawing development as it becomes more complex with each step, you’ll want to enlarge every picture. Note in the above picture that I have marked the crossing points of the trace and the plan lines for piece ‘b’ with small blue circles, numbered 1, 2 and 3. Note that point 2 is in fact crossing two lines which happen to be in the exact same place, as the center line represents the arris of the stick on top and on the bottom.

Now, I’ll zoom in a little closer on those crossing point, as I make a departure for new pastures:

You can see that I’ve extended lines from points 1, 2, and 3 up to the baseline of the elevation view of stick ‘b’. Like the lines which produced the elevation view of piece ‘b’ in the first place, these lines from points 1, 2, and 3 run at a 90˚ angle to the plan of piece ‘b’. Where these three lines run into the ground line for piece ‘b’, I have marked 3 new blue circles, denoted as 1′, 2′, and 3′.

Let’s pan back a little, with no changes to the drawing so you can be sure where we are:

It’s very important when you start adding and intersecting lines in these sort of descriptive geometry exercises to be crystal clear about each line, what it does and at which point it is intersecting and why. The three points, 1, 2, and 3, are the points of intersection between the plane, as it traces along the ground, and the plan lines for stick ‘b’. Our new points, 1′, 2′, and 3′, are formed as we transfer from the plan view of stick ‘b’ to the elevation view of stick ‘b’. My aim here is to produce a line on the elevation of stick ‘b’ which depicts that orange plane we saw a few pictures back, to show it crossing stick ‘b’ in other words. To produce that line, or any line, we need two points to connect. We have already produced three points, 1′, 2′, and 3′, however they are all at the same end so it is not those points we will be connecting. Not quite yet at least. We now need to produce a point on the other end of the plane. Having used the ground trace of the plane already, you might guess that we will now use the trace formed by the plane at the top of piece ‘a’.

We’ll start by first putting that orange plane, presto!, back into the scene:

Notice again the line which defines the top of the plane, and that it defines a cut line on the the top of the stick at the same time.

If I wanted to relate this upper line to the ground somehow, the most direct way would be to drop a pair of plumb lines down from the top corners of the orange plane:

You can see that where these two plumb lines meet the ground I have indicated it with large blue circles. Like a stone falling into a pond, plop!

Now, if we connect the two encircled blue points we just made, we would have this:

Here’s a view from a slightly different vantage point:

The main thing to understand here is that the line we just formed happens to run right through the white diamond defining the top cut of stick ‘b’. This line is the trace for the top of the plane, which is also the upper end of the stick.

Now, if you think back to the first few steps we took with the lower trace line in intersecting it with the plan lines for stick ‘b’, you might expect something similar, process-wise. Your assumption would be correct, however all I’m going to do is deal with one point of intersection between that upper trace and the plan lines of stick ‘b’:

If you look right in the middle of the above picture, following to the right along the trace line from the white diamond, you will see a small blue circle, marked 4, denoting the point of intersection. This is the point where the upper trace line meets the side arris of stick ‘b’.

Again, just as was done with the lower trace points of intersection, where we projected up at 90˚ from the axis of the stick in plan, to form points 1′, 2′, and 3′, we now project point 4 up 90˚ to form point 4′:

Now for the grand finale – we connect point 4′ at the top of the stick ‘b’ elevation view, with the point on the bottom of the elevation which corresponds with it. Point 4 and 4′ relate the top trace of the plane and the right side arris of the same stick in plan view. Which of the points on the bottom, out of 1′, 2′, and 3′ represents the same arris?

Let’s look again at the detail of the intersections down at the foot:

You can see clearly that it is point 3 which intersects the same arris as point 4 does, and therefore it is point 3′ which must connect with point 4′:

As I have written directly on the above drawing, the line formed between 3′ and 4′ defines that orange plane of stick ‘a’ as it slices through piece ‘b’ in elevation view. How about them apples?

Once we have established the line of the orange plane through stick ‘b’, we can add a few more lines. As the points 1′ and 2′ which projected parallel to the line 3~3′, they must continue in parallel to the line formed between 3′ and 4′:

Looking closely at the drawing, you can see that with these new projection lines, from points 1′, 2′, and 3′, we also find new points of intersection with the plan view, at 1″, 2″, and 3″. Recall that earlier in the post I mentioned that where point 2 was formed, at the place where the lower trace crosses the central plan line of the stick, that 2 arrises are represented by that plan line; in the elevation view development above, you can see that there are two points marked with 2″, one for each arris, the top and the bottom. The plan view’s central line also is representing two arrises, these two being exactly 90˚ rotated from the arris we crossed in plan at point 2.

Well, that’s quite enough excitement for one day. In the next post in this series we’ll look at what sort of useful fun we can have with the three lines we have just formed upon the elevation of stick ‘b’ in a drive to define the cut lines needed on stick ‘b’.

Thanks for dropping by today, and your outraged comments, or whimpers for mercy, as the case may be, are of course welcome as always. Hope you’re having fun with this drawing – I am!