Thirteenth and final post in this series, with previous installments found in the archive to the right of the page. In the previous post I completed the development of all the projection lines onto the unfolded view of piece 'b'. All that remains is to decide which lines are to be cut, and in many ways this is the hardest step in the entire process.
Here then is where we ended up in the last post, with the four planes of stick 'a' projected over onto the unfolded view of stick 'b' to give those 4 zig-zagging lines across the faces:
In the above drawing, several elements have been concealed to give a less cluttered view, including the elevation view of stick 'b'.
Now, to decide where to mark the cut lines we need to take a look over at the elevation view of stick 'b', where we had previously established the cutting planes. To refresh, here are the cutting planes for the 'front' and back faces of stick 'a' respectively:
And here are the 'left' and 'right' faces of stick 'a' forming their cutting planes on stick 'b''s elevation (note the outlines of the other two planes in relation to these ones):
In the next picture, I will place all four cutting planes in the scene at the same time, and remove the coloration from all planes so we only see their outlines:
Next I mark what is what in terms of the outlines of these planes and which parts of stick 'a' they represent:
Note carefully the places where one cutting plane intersects another -as i have indicated in the above drawing. A couple of those points are rather close together, so I provide a close up of that area:
Now I'll take one point of intersection between one plane and another and project it over to the unfolded view of stick 'b'. Of course, a point of intersection between planes is the same thing as a corner of the stick 'a':
See how the projection travels over and note its place of intersection - just where two of the lines on the unfolded view of 'b' intersect one another.
Continuing this process then, I find the following points of interest, by projection:
This really is the crux of it - you have to carefully consider where the corners between planes are formed, and, at the same time relate their positions to the various unfolded faces. One area that is likely to confuse is the connection between the arris at far left of the unfolded view with the arris at the far right. Remember that these lines are one and the same thing.
Okay, here then are the areas which need to be removed from the collision of the four plane and their lines projected over to the unfolded view of stick 'b' (drumroll please...):
I've also taken the liberty of shading the portion of the elevation view where material would need to be cut away.
Now, the proof lies in the pudding, so to speak. Since we have been working this past while on an unfolded view, a view that I showed I had developed by taking a virtual stick of wood and treating it as if it were a cardboard walled section and unfolded, then why not reverse the process?
Here's the fully unfolded view popped up off the 2D plan:
Now we start to fold it back together:
Folding is completed in the next picture:
We may as well swing the stick around into the position occupied by stick 'b' and place it at slope:
Next, I'll do a little virtual surgery and remove those shaded areas, then make some connections internally in the stick to create the section that is removed from stick 'b':
And now we return to the start, with both sticks combined, to show the fully resolved problem:
The unfolded view could be employed in a couple of ways to actually mark out stick 'b'. If the drawing were at 1:1 scale, we simply place the respective faces of the stick upon their corresponding unfolded faces on the paper/ground, and mark the spots on the arrises of the stick where the cutting plane lines cross through. This process is repeated for each face and then the tick marks at the arrises are connected across the faces, then one has to decide where to cut, and proceed from there. If the drawing is a scaled down, affair because one doesn't have the room for full scale work, then it would be a matter of measuring from point to point on the drawing, converting these measures by the scale, and then transferring to the stick. The angled lines across the unfolded view, including the top and bottom cut lines, could easily be transferred directly with the use of a bevel gauge (or three).
If one needed to detail different joinery, or mark the points of intersection of stick 'b' onto stick 'a', then the same process used to develop 'a' onto 'b' would be applied the other way. No need to detail that process here, as the technical methods are exactly the same as have already been described. Readers who wish to cement their understanding further may wish to do another round or two of this exercise, varying the plan angles of the sticks and seeing what kind of trouble they can get into. And it wouldn't hurt to cut a couple of sticks out to examine how the connection works as well.
I hope those readers that took the opportunity to follow along got something useful out of this drawing process. If I can obtain other useful drawing problems out of that same French text, and I get the sense that there is further interest in these sorts of explorations among readers, then we'll do another round like this later on in the year. Thanks for taking to time to involve yourself in carpentry study!
That my friends is that. Down to one build thread now, so it looks like i may have to introduce other topics soon. Stay tuned.