Tuesday, November 30, 2010

X Marks the Spot (IV)

This is the 4th post in a series describing a French carpentry drawing approach to solving a carpentry problem involving two crossed sticks. It's just a couple of sticks, how hard can it be? heh-heh!

For previous installments please refer to the archive at the right of the page.

Where we last left off I had walked you all through the steps required to draw the cross section of the stick on the floor. By simply swinging an arc down from the hypotenuse of the triangle which gives the slope of the stick, we were able to locate the end point of that square cross-section on the ground. From there we drew the rest of the square by simply projecting lines up that we 90˚ to one another. If you're feeling fuzzy about that detail, I suggest you read the previous post one more time.

Onward: now that we have the red cross-section in place on the floor, we can use it to draw a few other things. First up, we'll draw the lines for the other arrises of the stick upon the slope triangle. To do that, we start by taking the other corner points of the cross-section slice, our red square, and project from those points back to the baseline of the slope triangle, and from there, swing lines back up to the line which represents the cross section plane, like so:

Notice how these two lines are simply like return 'bounces' from the initial arc we swung down. It's just like a harmless game of tennis this drawing stuff.

Since the cross section is oriented in perfect alignment with the axis of the stick, both of those corners are on the exact same line - the centerline of the stick. So the middle line being swung up in the drawing in fact is representing two points which happen to be overlapping one another. If the cross section square was rotated a little one way or another, we would not have these lines overlapping and thus there would be 4 separate lines and arcs in play. We can save that kind of fun for another day.

Noting now the two red circles marked where our new arcs ran into the line representing the cross-section, we now project new lines from those red marks parallel to the hypotenuse of the slope triangle. Since the hypotenuse of the slope triangle is the same as the top-side arris of the stick, what we are drawing here are the other arrises of the stick:

Notice how those two new lines, parallel to the hypotenuse, meet the run of the triangle. Since that line giving the run is in fact the ground, the intersection of these two new lines with the run line is where the other arrises on the stick are meeting the ground.

Next, I'll transfer the lines from their intersections with the run line across, in a 90˚ orientation to the run, and at the same time extend lines down from the cross section. These lines are parallel to the axis of the stick, that is to say parallel with the run of the slope triangle. The meeting points are shown in the following drawing by the four red circles:

Those four red circles can now be connected:

What do we have there? It's the footprint of the stick on the ground. What we need now are some trumpets. Notice what has happened to our square cross-section (in red) when the slice across the stick is a horizontal one, not square. One of the diagonals of that cross section square has stretched some in the footprint.

Now, lest we neglect the upper end of the stick, we can now pencil in the lines for the cut on the top of the stick, which, if you recall, was done so as to make the top of the stick parallel with the ground (the same as the cut on the foot). So, it is a simple matter to extend a line parallel to the ground, which is our run line, and extend the other arris lines to meet it:

Next, we can do the same process we just went through, and transfer the points defining the top of the stick back down:

Continue these projections downward, and at the same time just as we did before, extend the lines from the cross-section view up to meet them. Where they meet will define 4 points of intersection, as it did with the foot, and we will produce now the view of the top of the stick, in white, sliced on the horizontal:

Pulling back to look at the big picture for a moment, what we have now drawn on the plan is a depiction of the stick, at slope, viewed from straight above:

In case your head is swimming at this point, or you feel a need to blink a lot or go for a walk, I'll put the 3D piece of wood back into the drawing and you can see how our 2D plan just completed nicely defines the various points of the 3D piece:

Since the stick was sitting right on the ground, in the next drawing I've moved it up a bit so you can confirm that all the points from the 2D are coinciding with the 3D stick:

And if I were to flip the stick over on it's side, much as we did a couple of posts back with the light brown slope triangle, we would see the following:

Amazingly, no animals were injured during the filming of this scene.

So, that concludes the basic layout of the that stick in 2D. The next step is to repeat the process with the other stick, something that dedicated readers might want to do on their own before the next post rolls down. You will find that lines will begin to progressively accumulate on the drawing, so watch out! Those of you familiar with using layers in your drawing work will want to keep the sketch of the second stick separate from the first. Have a go at that if you're feeling like it, and when this thread takes a stab at this problem next time, I'll quickly run through the drawing of that second stick in plan and then we'll start to look at the real problem at hand: the points of intersection where the sticks cross one another. Stay tuned, and thanks for coming by today. --> On to part 5


  1. "....the footprint of the stick on the ground". About thirty four years ago when carpentry work was hard to find I'd drive in from VA to Wash DC to the Library of Congress and read back copies of American Builder and Building Age and then go down the hill a bit to the Hdqtrs of the Carpentry and Joiners Union to read The Carpenter magazine - all late 19th and early 20th C stuff. There I first saw the drawings the 'footprint'. Long did I study these complexities of lines laid out on the dried up newspaper pages but to no avail. What delight it is after these many year to have that door so longer locked opened. Many thanks.

    1. Robert,

      always pleased to learn I have opened a door for someone - thanks!



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