This is the ninth post in a series describing the 2D layout of a problem of two crossing sticks, found in a 10-volume set on French Carpentry. For previous installments, please refer to the blog archive at the right of the page.
One point to make at the outset: if the reader hasn't been following along actively, trying to draw the problem along with me, it is more than likely that this series has long since moved into the zone of incomprehensibility. And today's post, though very much intended to aid in clarification, will only likely compound the confusion on the part of those readers who have not been following along with the process in an active manner.
Today's post is intended mainly as a consolidation of steps already achieved, hopefully clarification of what the lines already developed mean, and then small advance forward in the process.
Just to review the past few steps taken: we have extended planes from all four faces of stick 'a'. We'll consider them one by one. Here's what I'll call the 'front' face, giving the orange plane:
Here's the 'back' face, which is parallel to the front:
Now the 'left' side face:
And finally the 'right' side face, which is parallel to the left:
Each of those planes meets the ground, and leaves a trace line. That trace line runs over the lines indicating the plan of stick 'b' at slope; where those plane trace lines cross the plan, we develop points from which we make projections. By the time each of those projections has been done, we end up with a fair number of lines crossing the elevation view of stick 'b':
Now it is time to see what we can make of all those lines. Oh what a tangled web we weave when at first we...
Um - wrong literary quote for this situation!
(Sir Walter Scott , in Canto VI, Stanza 17 of "Marmion" (1808) an epic poem about the Battle of Flodden Field in 1513.)
Let's start by considering the very first set of projections we established, back in post 6, using the orange plane coming off the 'front' face. For clarity's sake I'll disappear the other projection lines for the time being:
To recap: the orange plane trace line on the ground ran into the plan lines of stick 'b', giving points 1, 2, and 3. Projections then developed, moving 90˚ to the axis of the plan, running into the ground line for the elevation view of stick 'b', and generating a second set of points, namely 1', 2', and 3'. From there we projected a line off the top cut profile of the stick (as this is the upper termination of the orange plane) and developed points 4 and then 4'. Connecting 3' and 4' gave us the line of the plane crossing the elevation view of stick 'b', and we then made lines parallel to line 3'~4', giving points 1", 2", and 3".
Now for some attempt at clarification. I contended that the line formed by connecting points 3' and 4' was the line of the orange plane on the elevation of stick 'b'. In case that assertion isn't entirely convincing to the reader, I thought that I could prove it by simply using a Sketchup feature and rotate that orange plane over and down until it cuts through the elevation view of stick 'b' - let's see if that maneuver gives us the same line drawn by the 2D method.
First off, we note that point 3 moved over 90˚ a certain amount to the side until it ran into the ground line for stick 'b', elevation view. This gave point 3'. The purpose of moving it over was so that the drawing of the elevation wouldn't be partially sitting atop the plan drawing, which would tend to make things more confusing I suspect. So, before we rotate the orange plane, we need to slide that plane sideways as well, the same distance and alignment from points 3 to 3':
Now the plane is in position, we rotate it. To do that in SketchUp, I find the point of intersection between stick 'b''s groundline and a plumb line from the top of the stick, and draw a small cube in that location. The cube is a contrivance upon which I can place the rotate tool to spin that orange plane over:
That cube is in white in the above drawing. Of course, if you're following along with this process with pen and paper, you can sit back and relax and just try to see what I'm trying to show. Hopefully it will be clearer than mud at least.
Now let's rotate the plane - first I'll turn it 30˚, an admittedly tentative start:
Then to 60˚ (getting more aggressive!):
And finally all the way over to 90˚:
Let's zoom in a little bit to confirm how this rotated orange plane relates to that line formed in 2D from point 3' to 4':
It would appear that the rotated plane indeed conforms perfectly to the line developed in 2D. Hopefully the reader is equally convinced, by this point and that the above exercise hasn't served to increase confusion in any manner. The 3D trickery has proved the 2D and now we can move on a bit.
Now, the entire point of doing a 2D drawing is to develop the cut lines on our sticks so that the woodwork can be done and the assembly realized. I've been using the 3D as a crutch in this drawing series to help clearly explain how the 2D works, and I'd like to continue in that vein.
Let's again look at the points and lines formed on the elevation view of stick 'b', simply by the projection of that front orange plane:
We note the points of intersection of the projection lines on the elevation: points 1", 2" (at two points) and 3". How do these relate to the cut lines we need?
Let's look at one of those cut lines on the 3D representation, specifically the one running from the top arris of stick 'b' to the left side arris:
Obviously, in order to cut stick 'b' to accommodate stick 'a' passing through it, we need to establish the cut line from the top arris to the left side arris on the stick.
With that in mind, let's look again then at the ground trace and see where it intersects the plan:
As the reader can see, point 1 is the intersection of the ground trace and the left side arris of stick 'b', while point 2 corresponds to both the top and bottom arris of stick 'b'. The line traced from the top arris to the left side arris in the 3D should also be doable in the 2D, no?
Let's swing aorund the view point to look at the projections from the trace onto the elevation view of stick 'b':
A little closer in:
Since point 2" (the one on the right side of the above drawing) is the top arris, and point 1" is the left side arris, we could draw a line connecting these two points, as shown.
We can do the exact same thing for another cut line - take a look aback to the 3D to see how this line travels from the right arris to the bottom arris:
Since that line from right arris to bottom arris is formed by the orange cutting plane, it would be parallel to the line from the top arris to the left arris, correct? Therefore, the line on the plan view representing that same connection would also travel parallel to the line we already drew, moving between the points of the plan representing the bottom arris (2") and the right side arris (3"):
So, 2"~3" is parallel to 1"~2".
Finally, we can complete the interrelation of points we saw in the 3D view, directly on the plan by connecting the top arris (2") with the right arris (3") and connecting the left arris (1") with the bottom arris (2"):
I'll highlight now the zone representing the cutting plane across stick 'b':
Of course, we can't simply just slice and dice down those cut lines quite yet, as they are unbounded by other cut lines, such as will be formed by developing the other planes from stick 'a' on the elevation of stick 'b'. If we cut the lines just drawn, we would slice stick 'b' into two parts, right where the orange plane, the 'front' face of stick 'a' cut across it. Not such a helpful step.
Next time I'll develop those other cut lines on the elevation in much the same manner as I have shown above for the orange plane. Motivated readers may wish to try this step for themselves to see if they grasp the concept. Hopefully my next post in this thread will confirm your work. Once the cut lines are developed, we will be ready to unfold the development of the sides of stick 'b', which would be the final stage in the drawing process before wood gets marked and cut.