Thursday, December 23, 2010

X Marks the Spot (VI)

Back at this gem of a French charpente drawing problem. This exercise was sent to me by a reader who in turn received it from his teacher at the American College of the Building Arts, Bruno Sutter. This is the 6th post in the series, with previous entries archived to the right side of the page.

Where we last left off, the work of marking out the basic configuration of the crossing sticks had been established in a 2D format:


Now the more challenging part of the drawing begins, which is the process of using the information we already have on our 2D to determine how the pieces intersect one another. At the end of this process we will have a drawing which we can directly measure and take angles off of, or, from which we could, if the drawing were done full scale, superimpose our pieces of wood upon the drawing and transfer the marks directly to the wood from the drawing to give our cut lines.

Let's just refresh the memory in terms of the basic layout issue here, with the 3D sticks -note that on one stick, piece 'a', I have colored the face we will work on today a different color than the rest, in the interest of clarity:


In the next drawing, I have taken that orange-colored face and extended it into a much bigger plane:

While the plane can run into infinity to the left and right, it is bounded at the bottom by the floor, where the stick terminates, and by the line of cut at the top of the stick. The line along the floor coincided of course with one side of the footprint of the stick, and the line at the top coincides with one edge of the top cut of the stick. Those lines, along the floor and along the top cut, are also parallel to one another and in complete alignment.

Next I have, presto!, disappeared the orange plane and left only the lines which defined it. We are most interested initially in the plane's trace line which runs along the floor:


Obviously, looking at the place where the sticks cross, you can see the lines of intersection formed between the orange face and two of the faces on piece 'b'. The line along the floor from the footprint of piece 'a' also crosses the plan view of piece 'b'.

Let's take a closer look at how that line, the lower trace from the orange plane, and observe how it intersects the lines defining the plan of stick 'b':


Remember that you can click on these pictures to make them larger- in fact, to have any hope of getting a handle on the drawing development as it becomes more complex with each step, you'll want to enlarge every picture. Note in the above picture that I have marked the crossing points of the trace and the plan lines for piece 'b' with small blue circles, numbered 1, 2 and 3. Note that point 2 is in fact crossing two lines which happen to be in the exact same place, as the center line represents the arris of the stick on top and on the bottom.

Now, I'll zoom in a little closer on those crossing point, as I make a departure for new pastures:


You can see that I've extended lines from points 1, 2, and 3 up to the baseline of the elevation view of stick 'b'. Like the lines which produced the elevation view of piece 'b' in the first place, these lines from points 1, 2, and 3 run at a 90˚ angle to the plan of piece 'b'. Where these three lines run into the ground line for piece 'b', I have marked 3 new blue circles, denoted as 1', 2', and 3'.

Let's pan back a little, with no changes to the drawing so you can be sure where we are:


It's very important when you start adding and intersecting lines in these sort of descriptive geometry exercises to be crystal clear about each line, what it does and at which point it is intersecting and why. The three points, 1, 2, and 3, are the points of intersection between the plane, as it traces along the ground, and the plan lines for stick 'b'. Our new points, 1', 2', and 3', are formed as we transfer from the plan view of stick 'b' to the elevation view of stick 'b'. My aim here is to produce a line on the elevation of stick 'b' which depicts that orange plane we saw a few pictures back, to show it crossing stick 'b' in other words. To produce that line, or any line, we need two points to connect. We have already produced three points, 1', 2', and 3', however they are all at the same end so it is not those points we will be connecting. Not quite yet at least. We now need to produce a point on the other end of the plane. Having used the ground trace of the plane already, you might guess that we will now use the trace formed by the plane at the top of piece 'a'.

We'll start by first putting that orange plane, presto!, back into the scene:


Notice again the line which defines the top of the plane, and that it defines a cut line on the the top of the stick at the same time.

If I wanted to relate this upper line to the ground somehow, the most direct way would be to drop a pair of plumb lines down from the top corners of the orange plane:


You can see that where these two plumb lines meet the ground I have indicated it with large blue circles. Like a stone falling into a pond, plop!

Now, if we connect the two encircled blue points we just made, we would have this:


Here's a view from a slightly different vantage point:


The main thing to understand here is that the line we just formed happens to run right through the white diamond defining the top cut of stick 'b'. This line is the trace for the top of the plane, which is also the upper end of the stick.

Now, if you think back to the first few steps we took with the lower trace line in intersecting it with the plan lines for stick 'b', you might expect something similar, process-wise. Your assumption would be correct, however all I'm going to do is deal with one point of intersection between that upper trace and the plan lines of stick 'b':


If you look right in the middle of the above picture, following to the right along the trace line from the white diamond, you will see a small blue circle, marked 4, denoting the point of intersection. This is the point where the upper trace line meets the side arris of stick 'b'.

Again, just as was done with the lower trace points of intersection, where we projected up at 90˚ from the axis of the stick in plan, to form points 1', 2', and 3', we now project point 4 up 90˚ to form point 4':


Now for the grand finale - we connect point 4' at the top of the stick 'b' elevation view, with the point on the bottom of the elevation which corresponds with it. Point 4 and 4' relate the top trace of the plane and the right side arris of the same stick in plan view. Which of the points on the bottom, out of 1', 2', and 3' represents the same arris?

Let's look again at the detail of the intersections down at the foot:


You can see clearly that it is point 3 which intersects the same arris as point 4 does, and therefore it is point 3' which must connect with point 4':


As I have written directly on the above drawing, the line formed between 3' and 4' defines that orange plane of stick 'a' as it slices through piece 'b' in elevation view. How about them apples?

Once we have established the line of the orange plane through stick 'b', we can add a few more lines. As the points 1' and 2' which projected parallel to the line 3~3', they must continue in parallel to the line formed between 3' and 4':


Looking closely at the drawing, you can see that with these new projection lines, from points 1', 2', and 3', we also find new points of intersection with the plan view, at 1", 2", and 3". Recall that earlier in the post I mentioned that where point 2 was formed, at the place where the lower trace crosses the central plan line of the stick, that 2 arrises are represented by that plan line; in the elevation view development above, you can see that there are two points marked with 2", one for each arris, the top and the bottom. The plan view's central line also is representing two arrises, these two being exactly 90˚ rotated from the arris we crossed in plan at point 2.

Well, that's quite enough excitement for one day. In the next post in this series we'll look at what sort of useful fun we can have with the three lines we have just formed upon the elevation of stick 'b' in a drive to define the cut lines needed on stick 'b'.

Thanks for dropping by today, and your outraged comments, or whimpers for mercy, as the case may be, are of course welcome as always. Hope you're having fun with this drawing - I am!

9 comments:

  1. I thought I had it all done, but then I noticed differences with the example. So I started the drawing all over.
    But that's not the problem, fighting Sketchup, while letting it do drawings it was not made for, is more painful. Mainly fighting the Tape Measure Tool, as sometimes I get an (infinite) Construction Line and sometimes a Construction Point. Adding a point script did not help as I also need a construction line between the original point and its projected point to document its meaning. The lack of Contruction Arcs is another annoyance.
    But OK, the use of Inferance in Sketchup is great, and drawing all this by hand would be slower and give major precision problems.

    ReplyDelete
  2. Damien,

    yes, there are plenty of pitfalls on SketchUp for the unwary in regards to doing accurate 2D drawings- I manage to find these pitfalls on a regular basis myself. How many times, I wonder, can I step in the same hole?

    The example you mention of the segmented arcs that SketchUp does are a case in point. An easy place to introduce slight errors into the drawing, especially given the lack of ability to attach a line to the tangent of an arc or circle (unless you obtain a plug-in).

    That said, I intended this series to be less about SketchUp and more about the drawing progression, which could be easily done with pencil, compass, and straightedge. I think SketchUp might help in making the explanation of the 2D drawing a little clearer, and that is the main reason I am using it to do the drawing. Probably you well realize all that, nonetheless for benefit of other readers, I felt that I should reiterate the point.

    Cheers,

    Chris

    ReplyDelete
  3. Chris,
    I am boggling with admiration about your blog. I have been dipping in here and there over the last few weeks.
    This series particularly interests me because I hope it will show me a graphical way to solve the top-cut angle of a hip rafter.
    Like stick "a", twisted so that the footprint is a rectangle, rising to meet a vertical plane at an oblique angle.
    Any hints ?!
    Greetings,
    Robert
    PS A bit like Damien, I cannot find a way reproduce your arcs easily. A "compass tool" would be a great SketchUp addition...

    ReplyDelete
  4. Robert,

    if by, "a graphical way to solve the top-cut angle of a hip rafter", you are referring to the backing cut, then this particular exercise will not be of help. I haven't done a post specifically on backing cuts yet, though it will happen sometime soon I'm sure. In the meantime, perhaps you might want to glance at the following post, which covers some of the possibilities in that regard:


    http://thecarpentryway.blogspot.com/2010/03/following-mazerolle-theorie-des-devers.html

    I'll address your comment about SU and the drawing of arcs in the next post.

    ~Chris

    ReplyDelete
  5. Chris,
    Sorry, used wrong terminology: I meant side-cut, not top-cut (but why side-cut?). I would like to measure off the angle across the top surface of the unbacked hip, between the top of the hip and the ridge (or kingpost in your example). I believe this is Hawkindale angle R4, except the top surface instead of bottom surface of hip - if that makes a difference.
    Greetings, Rob

    ReplyDelete
  6. Because I am using a drafting board (pencil, compass and eraser etc) I have found tha making models of sticks A and B was helpful. These I did with 3/4 stock having laid out for that size from the start . I cut these in a wooden miter box with a 45 degree chamfer in the corner so to hold the aris in plumb position and cut on the pitch (38.7 degrees for A and 30.5 degrees for B. specially helpful was using 45 degree 3/4x3/4 chamfered stick to lay down on A and B in their 2D elevation. Now with my models I am able to track down all the little dihedral s in vivo! Great fun Thank you again

    ReplyDelete
    Replies
    1. Robert,

      excellent that you have taken the matter into your own hands and made a model- well done!!

      ~C

      Delete
  7. Going over theses sequences I believe that drawing 11 and 12 are reversed. What is now 11 should be in position 12 and vice versa

    ReplyDelete
  8. Robert,

    it's simply the case that I took a drawing with some later developments on it - namely the line from 4 to 4' - and placed it in as drawing 11. The subject of drawing 11 however was the trace of the plane, not line 4~4'. I can see why that would have been a little confusing.

    ~C

    ReplyDelete

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