Saturday, July 16, 2016

La Menuiserie Study (II)

Post 2 of a series.

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After working my way through the first 70 pages of the French text La Menuiserie, I at last arrived at the keenly-anticipated first meaty exercise to be detailed, on Plate 1, entitled, "Pénétration biase de deux parallélépipèdes rectangles":


The term 'parallelepiped' is a word which, though seldom encountered by most I would suspect, is a part of the English language. While the term describes a solid comprised of 6 faces which are parallelograms, when you recall that both a square and rectangle are a form of parallelogram (i.e., are 4-sided figures with two pairs of parallel sides), you can see it is a therefore a term with which one can describe a S4S stick of wood. Try throwing that term out to your co-workers on the job site one day and see what sort of response you get - hah! It's a term you'll come across in some English 19th century woodworking texts.

The problem is one of a square section vertical post being pierced at an angle by a smaller square-section stick. Note that the smaller stick crosses the post at a given angle, and is not centered to the post, and is itself rotated to a given angle. They are not looking to provide an easily-solved example. The purpose of the Plate 1 study is shown to the right of the sketch above, the développement, which is an unfolded view of the four faces of the post, showing the position and shape of the mortise lines.

I was psyched to dig into this first plate in the text. I cranked the hand wheel, the light's flickered, a screech of a leather belt and SketchUp came to life. I faithfully drew out the plate in 2D and produced the required unfolded, developed view of the mortises. It seemed straightforward enough to follow the method shown.

Curious to check the result, I then flipped the unfolded view upright, and proceeded to fold the faces inward until I produced a '3D' post. Then I connected the mortise corner lines together within that post. All seemed well as planes were formed within. Then I drew up a second smaller stick of the designed dimension, rotated and tilted it to the correct position and placed it into the through mortise. Jolly nice and all, except it was not even close to a decent fit. Not only was the mortise itself non-square internally, wall to wall, but it was not square dimensionally either.

Huh? I figured I must have blundered somewhere so I drew the entire exercise again, only to reach the identical outcome. Something was haywire. I then decided to do a reverse engineering, making a post without any mortise layout, using the plan and elevation views to place the smaller crossing stick in the correct position and then intersecting the two 3D parts so as to produce the mortise lines on the post. Then I unfolded that post section and laid it flat atop the drawing to compare, thinking that maybe I had simply put one of the mortise corner intersection points in the wrong place or something like that. Here's the overlay:


As you can see, while the overall mortise outlines are similar in overall appearance, they have only one point in common, at the top, at each side. The lines associating to the white background are in the correct places, while the colored pieces in the background are part of the development done from the original drawing.

The text had to be wrong somehow, as the 3D did not lie, and later that evening, while pacing and bouncing with the baby on my shoulder, I could see what the issue was: something was causing the layout of the mortises on the post to be distorted, and I had an idea as to what the culprit was.

The following morning, when I had a little time, I went back to the sketch and tested out my theory, and then was able to confirm my suspicion. Here's the problem area of Plate 1:


The square labeled Se1' through Se4' is the cross section of the sloped crossing stick. From the four corners, lines are projected plumb down to produce Report Line 'A'.

Report line 'A' is used to draw the plan view of the stick:


I suspected that the way Report Line 'A' was established was the kernel of the problem - that report card gets an 'F' unfortunately.

I placed the 3D stick directly over the plan view to check whether the plan of the stick conformed to the actual outline of the stick - it clearly did not:


A closer look with a couple of width measurements indicated for comparison:


A roughly 4mm difference in plan view width, not to mention the discordant positions of the arris lines, would definitely account for distorted mortises in the development of the mortises on the unfolded post faces.

Just to clarify, the actual crossing piece section conforms perfectly with the elevation view on the drawing, as the traces dropped down to the floor indicate, and is the exact size of the cross section depicted on the elevation view:


Anyhow, shortly afterwards I sorted out the problem with the text's drawing. The projection from the crossing stick's cross-section should not drop down plumb, but project square to the stick, like this:


I then swung the lines down to create a similar looking horizontal Report Line 'A' as the text had shown. These lines could be further extended, if desired, and reflected on a 45˚ line to directly produce the plan view lines for the piece.

With the revision to the report line in place, I re-drew the plan view of the sloped crossing stick, and from there drew out the development one more time:


The 3D stick was then unfolded, and was placed atop the drawing development for a look-see. The mortise outlines on the development and the unfolded stick conformed to one another exactly.

Another view of the post and post assembly with sloped piercing cross piece:


So that was not a promising start to the eagerly-awaited 'meat' of Volume 3 of La Menuiserie. Planche 1, presumably the most basic exercise, had a serious error.

Well, in truth the method shown was correct save for that one issue (projecting the lines for the 'Report Line 'A'), however that one little mistake threw the development out by more than a little. I'm puzzled how this could have gotten past the editors of the book. Did no one actually make the project? The drawing in the book looks to be produced in CAD, so it seems like they never modeled it in 3D. Or was it so basic a model that a lazy or simple mistake was overlooked? Who knows....

As things have turned out, over the subsequent 7 plates which I have completed, all dealing with the lines of intersection between solids, have all been spot-on, so I think that first plate has an unfortunate error in an otherwise excellent text.

Thanks for dropping by and stay tuned for part III.

8 comments:

  1. Exercises like these are really meant to be done on a drawing table, I think.
    Doing them that way would give a much better "feel" for the problem.

    If I had to to this, I would probably start with the center lines of the beams in side view. Then add the cross-sections of the respective beams on the center lines and proceed from there.

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    1. Roland,

      I'm sure they are meant for drawing on paper, and the means of proceeding you suggest is fine. I followed the text to the letter, starting with a plan view of the post.

      It is true that pencil and compass on paper has a certain feel, however if I had drawn it that way, and not actually laid out on wood and cut the pieces afterward (i.e., trusting completely in the text), I may not have discovered the error in the drawing. Having the ease of converting parts to 3D solids and confirming the 2D is a decided advantage I think, and no wood needs to be wasted. Also, some problems are easier to understand in 3D presentation - like cutting planes and their relationship to the parts on the floor.

      ~C

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    2. Hello Chris,
      When I look at your result, it is as if you don't use the same angle of rotation(around the long axis) of the oblic piece. It seems also that a little translation will also make the book and you to be in accordance.Try to rotate it 5° to the right (when you look from left to right) and tranlate 2mm to the ..... good direction.
      This kind of difference is very sensitive to these initial conditions (butterfly effect!). Have you got the data from the drawing or from the text. The relative position of the two parts comes from the drawing (generally in this kind of book) So it is difficult to reproduce it exactly as no information is given.
      Phil

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    3. Phil,

      appreciate the comment. It shouldn't matter at all what the angle of rotation is, or the slope of the stick - either the geometrical drawing method works, or it does not. I've translated the text, and it makes no mention of a specific amount of rotation or a specific amount of slope for the piercing stick. If a specific slope or rotation were important, that would be mentioned or depicted. There is no 'butterfly effect' in this type of situation. Different slopes or rotations of the piercing stick will lead to different shapes and positions of mortises on the developed view, however, those variances will still lead to an outcome where the mortise allows for a square section stick of the correct size to pass through the post. With any variance in the axial rotation of the sloped stick, Report Line 'A' dimensions will also vary.

      Though there are no dimensions or angles provided on the text or plate 1 drawing, I do try to duplicate the drawing in the text as closely as I can, in the full light of the fact that it need not be a carbon copy. Thus, in my sketch, I still have the general arrangement of the sloped sticks arrises in the same relation to the post as the text does.

      Let's look at this idea a little more though, and see what might happen if I set the axial rotation of the piercing stick differently. If I rotated the piercing stick's section so that two of its faces were parallel to the slope, and two were therefore perpendicular to the slope, then logically the sloped stick would have its side faces plumb. That means that the plan drawing would present a print which was the same width as the stick, correct? However, if you consider how they project lines plumb from the sloped stick's section to form the Report Line 'A'', then it is immediately apparent that Report Line 'A' would be way too wide in such a case.

      In fact, the only case in which dropping plumb lines down from the arrises of the sloped stick section to make Report Line 'A' would be correct, would be where the stick were not sloped at all, but were horizontal. That would be the case where the lines are both plumb and are perpendicular to the slope of the stick.

      ~C

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  2. Out of curiosity, are the "pro" features in SketchUp needed to do this work, or does the free version have all of the functionality?

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    1. Hi Jamie,

      thanks for the question. The free version has all of the functionality required for the sketches you see here. I have the pro version, however truth be told I rarely make use of the features for which you pay extra, like Layout.

      SketchUp is easy enough to learn how to use if you are starting from ground zero, and there are plenty of online (Youtube) tutorials you can check out to learn about various functions. Of the many functions built into Sketchup, I only use a portion myself, remaining a bit ignorant of the rest.

      ~C

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  3. How do you handle Sketchup's segmented circles with the developed drawings?

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    1. Hi Brad,

      you know the weak spot of SketchUp as well as I do, eh?

      If you set the circle's facets to too low a number, it distorts the positions of the intersections too much. If you set it to too high a number of segments, then intersection of solids or components will not work properly a lot of times.

      I've settled on using 'circles' with 96 sides, as this seems to provide adequate accuracy, and no problem when intersecting parts.

      When I am taking a point off of a circle, I can zoom in and see whether I am meeting the middle of a segment (which is slightly less than the radius) or the arris of two segments meeting (which is the exact radius). If it is the middle of a segment, I draw a radius line through the point at the correct dimension, and go off of the tip of the radius line instead of the circle segment, if you follow.

      I've found the faceted circles mean that total precision cannot be achieve, but you can tell all the same if the geometry is correct or not. If a given intersection of line and circle is off by some tiny amount from a projection line (like 0.001mm) , then you can put it down to the issue of the segmented circle and generally have faith that the intersection is where it should be. If the line of projection is significantly off the line of the circle (by more than 0.1mm, say), then the procedure for the development/projection of the line is going to be at fault.

      If I am swinging an arc, and know the radius of that arc (which easy to measure in SketchUp), then I can first draw the projection line entering and exiting from that arc at the correct distance from the center of arc, and then draw the arc in afterwards, in which case the arc becomes more symbolic than an actual reference for a distance.

      In any case, even with a circle with 96 segments and the inherent uncertainty that brings, I think that Sketchup offers greater precision, with the capacity to zoom right in, than would pencil, compass and paper, using your unaided eye.

      ~C

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