Post II in what is planned to be a relatively short carpentry drawing thread, detailing a French carpentry problem and its 2D descriptive geometrical solution.
Here’s our problem, a situation in which 2 sticks are intersecting in the rough form of an ‘X’:
These sticks have different slopes from one another and are oriented differently to the floor plan from one another.
Let’s give this problem some parameters, so that those who wish to try this at home can follow along more easily. Taking the left side member, which is colored light brown, in the next illustration I show some of it’s particulars:
This piece is rotated in plan 60˚ from a line which connects our two sticks on the ground. I have labeled this piece as ‘A’ where it meets the ground, and it will be called piece ‘a’ from here on out. Notice that if you follow the arris of the stick from the ground, at ‘A’, up to the top, we reach a point marked with a small circle and which has a line running vertically through it down to the ground. This line is plumb to the floor and meets the floor 135 cm away from line A-B.
Point ‘B’ on the line, which is located 150 cm from ‘A’, defines the place at which the second piece, colored yellow, meets the floor:
This is therefore piece ‘b’, and it runs 45˚ to the plan. Again, following the arris of the stick from the ground to the top of the stick, we find a small circle and a plumb line back down to the ground. That plumb line meets the ground 150 cm from the A~B line.
Here are both sticks ‘a’ and ‘b’, shown in their positions relative to one another:
Note that the height of the two circles at the top of each stick is exactly the same for each piece. I chose 125 cm for that height, but it could of course be whatever height it needed to be.
Now let’s translate the previous 3D drawing into 2D. What we want to create first is the basic ground plan. Here’s what I come up with:
Hopefully that was straightforward enough for everyone to follow. If not, carfully compare the above sketch to the one previous, and you should see the common points.
In the next illustration, I bring in a little 3D once again, superimposing it directly on that 2D plan. I draw in a triangle representing piece ‘b’, from the point it meets the floor at ‘B’, up along its arris, to the circle at the top. This forms a triangle which represents the slope of that stick of wood:
That right-angled triangle has a rise of 125cm and a run of 150cm x √2, which is about 212.132cm. It’s not critical to know or understand why the run of the triangle is √2 times longer than the distance A~B. Those who have obtained my first two volumes of the carpentry drawing essay series should be quite savvy about this, but for completing this particular drawing problem it is not too important at all. Don’t sweat it if that √2 thing isn’t making sense.
Now let’s look at the triangle which defines the slope of piece ‘a’, which I’ll just add on to the previous sketch:
Again, piece ‘b’ runs along 45˚ to the line on the floor A~B, while piece ‘b’ sets off at a 60˚ from that same line. Both triangles rise the same amount, 125 cm.
Of course we are looking at a 3D illustration at this point, and, to follow traditional carpentry practice what we want to work with is a 2D drawing. You need to be able to visualize the 3D and translate it into 2D – that’s the key to this sort of drawing work, and it by no means comes easy, especially as the constructions gain complexity.
Those triangles which are sticking up in the 3D drawing need to be placed on the floor somehow. What we do is to treat those triangles as if they were a folded-up flap of paper attached to the ground. We simply fold them down, using the run of the triangle as if it is a long hinge. Let’s do just that with the light brown triangle representing piece ‘a’:
The illustration attempts to show, through the use of the multiple triangles, how the triangle for the slope of piece ‘a’ is rotated from a plumb position down to the ground. Maybe one day I’ll get all fancy and do some sort of animation for this, but for now the above will have to suffice I’m afraid. When the triangle is down, it would look like this in our 2D plan:
Note that nothing has changed in regards to the geometry of this triangle – it still has a height of 125cm, only now we’ve drawn that on the floor directly, and at a 90˚ angle to the run of the triangle. We’ll be doing the exact same thing with the other triangle which represents the slope of piece ‘b’ too, however we’ll stop here for today. We’ll do some more work on this light brown triangle next time. If you are finding the whole ‘rotate the triangle’ part of this a bit mystifying, I would suggest making a cardstock triangle to represent the slope of piece ‘a’ and rotate it on the drawn plan just as I have illustrated. It’s a simple 90˚ rotation downward. Up down, up down – it will be clear after staring at it long enough I hope.
Thanks for coming by the carpentry way today. –> on to post 3