X Marks the Spot (III)

A short thread on describing a French carpentry drawing technique. This is post 3, with previous installments located in the 'blog archive' to the right of the page.

We left off last time having formed a right angled triangle defining the slope of one of our pieces of wood. Piece 'a' to be precise. Here is the problem we are working with, showing the two sticks of wood and their respective slope triangles:


To be clear about this, I'm only showing the 3D parts here so as to aid in understanding the 2D. I am actually trying to teach the 2D only, and the 3D is merely helpful in that regard. There is no need for the reader to draw in 3D - all of this can be accomplished with pencil, straightedge and paper. Oh, and a compass too. I suggest that readers who are working along a drawing of their own along with these posts make a point of reading all the way through the post to the end before starting to sketch.

Today we will continue on with working on piece 'a' - here it is in all its glory:


I placed the slope triangle right inside the piece of wood, coloring it white in this instance so it is distinct from the piece of wood. Notice how the arris of the stick is coincident with the hypotenuse of the slope triangle.

And now, here's that slope triangle, which I have colored light brown again, rotated down to the floor, which was the last point where we left off in the previous posting:


If the step of rotating the triangle down is causing you consternation, please take a look again at post 2 in this thread.

Now, the next step in our 2D drawing is to draw the cross section of our stick. The view of the cross section of the stick will help us accomplish further steps in the drawing. A cross-section is taken at a 90˚ slice to the principal axis of our stick. The prinicpal axis of our stick is along the arris, which also happens to be the hypotenuse of the slope triangle in the picture above.

I'll just return to a 3D view for a moment so as to be crystal clear (I hope!) about what constitutes a cross section slice of our stick. I just said that a cross section is 90˚ to the long axis of the piece, which is the arris and the hypotenuse of the slope triangle. In the next picture, I have drawn in a plane which is oriented 90˚ to the slope triangle's hypotenuse:


That cross-cut plane, you might notice, intersects the floor right at the point where the slope triangle sun and rise meet. It doesn't have to meet there, as I could move that 90˚ slice up or down along the arris/hypotenuse as I might like. Placing the plane where I did however makes it clear that the cutting plane is very much like the page of an opened book, a page hinged along the line it makes at the ground. It's a convenient place to put the plane.

Next, I've placed the stick back in place so you can see clearly how the cross-section plane and the piece of wood relate to one another:


Now, if I were to delineate the position of the stick which is cut through by that cross-section plane, we would have the following illustration:


The red diamond shown above, which is actually a rotated square since our piece of wood is square in section, is the actual cross section of the stick.

Now, that's all very nice, but as mentioned last time, we seek to represent this cross-section in a 2D drawing, not a 3D drawing. We need some way of rotating that red section view down to the floor.

For starters, I'll take the stick of wood out of the picture so we can consider this in a clutter-free manner:


The red cross-section is part of the larger cross-section plane, a plane which attaches to/passes through the ground plane. If we treat that line along the floor formed where the cross section plane runs into the floor as a hinge, we can see that the entire plane may be swung down to the ground, just as if you were folding a page of a book flat:


Now I'll put that stick of wood back in their to clarify that all I have done here is swing the stick's cross section down to the floor:


That's doesn't seem too hard to grasp now does it? Let's look now at how we would accomplish that swinging down of the cross section in a 2d manner. First of all, I mark the line for the cross section plane onto my light brown slope triangle, a line that is 90˚ to the hypotenuse:


Imagine in the above picture how the same step looked in the 3d depiction. Now here's the 'tough' bit - we rotate that 90˚ line down to the ground, the ground of course being synonymous with the run of the triangle:


The red circle indicates the spot where the top of the cross section meets the floor. If you find this last move confusing, it might help to look at again at the same process as it occurred in the previous 3D renderings.

Now that we have marked the spot where the top of the cross section meets the floor, we can illustrate the cross section itself. The cross section of our stick is a square, oriented so that the upper and lower arris are in a plumb relationship. We can project off from the ground, at the point where our cross-section arc meets, a pair of lines:


These two lines are 90˚ in relation to one another, and are 45˚/135˚ to the line giving the run of the triangle. I won't go into how to lay off those angled lines, but it will be a simple matter to accomplish with a framing square or a commonly available plastic 45˚ drafting triangle.

Next we decide what the measurement of the side of the stick is going to be. I've chose it to be 20cm in my drawing, so let's mark off that measure along each side, and complete the drawing of our cross section:


Again, it might help to re-imagine the same steps that produced the above drawing as being a representation of the process in 3D.

That completes the drawing of the cross section view of the stick, which in turn, in our next installment, will allow us to draw a few more things. One thing though - you can see how the cross section view in red and the slope triangle in light brown are kind of on top of one another. We can de-clutter this part of the drawing by the simple expedient of sliding the light brown triangle off to one side, like this:


You can see that I've slid the triangle along it's rise, moving it far enough to be completely apart from the cross section picture. Notice how the mark from the arc we swung across that light brown triangle is connected now by a straight line to the top of the cross section. Further, this line is 90˚ to the run of the triangle, as are the lines you can see connecting the lower tip of the triangle and the lower 90˚ corner of the triangle. If you imagine that light brown triangle having little nails in those spots along its run, you can see that dragging the triangle off to one side would leave a trial of marks connecting it to the line where it once was. I hope that makes sense!

The whole point of moving that triangle over is simply for clarity, as will become apparent in subsequent posts. If you are doing this drawing on the computer, moving the triangle will be a simple task. If pencil and paper is your choice, then it would be a good idea to illustrate that triangle off to one side at the get-go of the steps shown today. If I had moved the triangle over at the outset, I suspect it might have confused people.

Thanks for coming by the Carpentry Way today and if any questions come to mind, feel free to share them.

On to post 4