Saturday, January 15, 2011

X Marks the Spot (X)

Tenth post in this thread on the topic of French carpentry drawing, showing how descriptive geometry in 2D is used to develop the cut lines on timber. In this case the problem is one of two sticks, one rotated 45˚ to the plan, and one rotated 60˚ to the plan, which cross one another and intersect faces. Previous posts in this topic are found in the blog archive to the right of the page, and those readers interested in French carpentry drawing in general and who are more or less new to this blog may wish to search for more in the Labels Index to the right of the page, under "Louis Mazerolle", and "French Carpentry Drawing". I've been exploring French layout in some detail for the past few years, and yes, I've nearly lost a grip on my senses. Yet, I forge onwards....

Where we last left off, I spend some time clarifying and consolidating the material from previous posts and then went on to show the points of connection between some of the points found in the first projection (the orange plane), originally explored in post VI of this series. Here's how that looked:


The points we connected on the elevation view I shaded in the above drawing in pink.

Now, since those points connect to one another, they should form a plane if we develop them in a vertical direction. So, just to show how that looks, I'll bring a bit of 3D magic back into the scene. I'll take piece 'b' and rotate it down:


As in the previous post, in SketchUp I create a little white cube upon which I can employ the 'rotate' tool and simply swing the stick 90˚ over to the left.

In the next drawing, I have taken the rotated stick 'b' and placed it directly above it's own 2D elevation view:


Now I project lines from the pink shaded area on the elevation, vertically (that is, plumb) upward and intersect those lines with the arrises on the hovering 3D stick:


As expected, a plane is formed within the stick, shown in purple in the above drawing.

Here's another view with some of the clutter removed:


Now the problem is that we cannot make much use of the above orientation of stick 'b' if we wanted to transfer lines right off of the 2D elevation and onto the stick. It is not convenient because the stick is rotated 45˚ axially. Too awkward to work with. What we need is a way where the stick can lay on one of its faces.

In the next drawing I have placed a generic stick above the drawing, a stick which has its lower and upper face normal to the floor:


What we want to do is represent that stick in 2D, however if we projected plumb lines off the stick back down onto the ground all we would show would be one face of the stick. What we need to do is unfold the faces of the stick, almost as if the stick were nothing more than a cardboard hollow shell.

So, let's do that - I begin the unfolding:


Next:


We can ignore the end caps of the stick as we are going to be cutting the ends of the stick off at a different angle. With the cardboard all flattened out, we have the following unfolded plan in white:


The above drawing shows how we would develop the faces of the stick in 2D; coming 90˚ off of the elevation view, we offset a certain amount (I chose 30 cm), and then measure off 2o cm and mark a line, repeating the width for each face, 4 times. These 4 lines of course are parallel to the line of the stick's elevation.

Next, I will select which of the 4 lines of the unfolded view will be made to correspond to the top side arris of stick 'b' - it could be any line really, however I have chosen to nominate the second line over as being the correspondent of the top arris of stick 'b':


With the top arris and corner 'B' established, I can connect some points to form the cut lines for the foot cut on stick 'b'. I first will mark out the left side arris of stick 'b' where it meets the ground, a point I denote as C. The line from C projects over to the unfolded plan view to give point C':


Since B connects to C at the foot of stick 'b', it follows that B' and C' should connect to one another on the unfolded plan view:


Now, in the elevation view, as I have noted several times in this thread, the left and right side arrises overlap one another. The left side arris I have already called C, the right side I will call D, and therefore all I need to do is continue the projection line just done previously over a little further to the right to give point D' on the unfolded plan:


Point D' is to the right of C', much as D is right of C. Again, we can connect points C' to D' on the unfolded view given that the points C and D are connected to one another at the foot of stick 'b'.

Then I can deal with the final arris, the bottom arris, which I will call point E at the place where stick 'b' meets the floor, and then projecting over from E over onto the unfolded plan to give point E':


It is important to realize one curious point about the unfolded view: the arris line on the far left and the line on the far right are actually one and the same. Therefore, the point C' already marked in fact is found on two points on the unfolded plan. Establish this second point C' and then connect a line over to it from E', just as E and C are connected on the foot of stick 'b':


If I trim the excess off, as it were, I reveal the lines for the end cut of our stick:


If one placed an actual stick directly upon the unfolded plan, it would be a simple matter of placing the appropriate face upon each unfolded face and marking the spots where the zig-zagging cut line meets each respective arris line.

Now we can move to developing the pink shaded are on the elevation onto the unfolded view. First, I'll project over, at 90˚ of course, the points for the top arris and the bottom arris:


The next step is slightly tricky as you have to be clear on which side arris is left and which is right, and correspond those points correctly. Here is my best guess:


Well, it wasn't actually a guess. Hopefully the reader is following along and hasn't either fallen asleep or is furrowing their brow. If you did them the other way around, you will find yourself presented with a curious zig-zagging cut line when you go to connect them in the next step.

With the points from the pink shaded area transferred over, all we have left is a simple connect the dots - be sure to carefully consider which points to connect to one another:


I followed that move by transferring the lines for the top cut of the stick, in much the same manner as has already been described above. Now we are at that happy point where a stick of wood with sides of identical width to the unfolded plan view could readily be marked out for top and bottom cuts, as well as the cut lines for that front face of stick 'a'.

We have followed through quite a ways now, and developed that front face, the orange plane, of stick 'a', onto stick 'b', then developed an unfolded view of stick 'b' and transferred all the lines associated to the orange plane's ground line trace intersection with stick 'b''s plan view.

Lines are mounting and massing on the drawing, as I'm sure the reader will have noticed, however we are closing in on the goal now. In the next post in this series I will work on developing the cut lines for the other faces of stick 'a' upon stick 'b'. I hope you'll stay tuned.

Thanks for your visit today. Comments always welcome.

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