tag:blogger.com,1999:blog-19304790559844185452024-03-13T19:59:32.067+01:00StereotomyTim Moorehttp://www.blogger.com/profile/09629429704217731021noreply@blogger.comBlogger10125tag:blogger.com,1999:blog-1930479055984418545.post-72802729045106307332012-04-27T09:01:00.000+02:002012-04-27T19:45:39.303+02:00LiningThis next post was supposed was supposed to be on aligning timbers before performing the operations described in our initial look at <a href="http://stereotomy-blog.blogspot.com/2012/03/le-piquage-or-theory-of-french-scribing.html">the French scribe method</a>. However, I have had the good fortune to be given a presentation on lining by Jean Pierre Bourcier, and I've decided to translate it from French and present it here. Lining is the preliminary step of marking reference lines on timbers. In particular, the reference lines are projections of an axis, used to align the timber on top of the full scale drawing, onto the faces of the timber. It works with irregular timbers and is a prerequisite to scribing. Without further ado, here is Jean Pierre's presentation, with a couple of my comments. If any reader spots phrases that could be better translated into specific timber framing terminology, please let me know in the comments.
<h3>Lining</h3>
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Preliminary operation for using the <i>niveau de dévers</i> method with the ground plan.
<p>
<h3>Definitions</h3>
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<ul>
<li>Lining: tracing, on two adjacent faces of a irregular timber, the projections of a line chosen to be the timber's axis.
<li>Counter lining: projecting those lines onto the two opposing faces of the timber.
</ul>
<p>
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<ul>
<li> Square line (<i>trait carré</i>): line perpendicular to a given line.
<li> <i>pièce de niveau</i>: timber that has been leveled along its length.
<li> <i>pièce de dévers</i>: timber that has been leveled across its width i.e., has no twist.
<li> Reference area (<i>plumée</i>): small rectangular area on a surface which has been dressed (planed) smooth, <i>[ so that a level can be accurately placed on it]</i>. This serves as a reference for lining and counter lining.
</ul>
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<ul>
<li> to plumb: to trace a vertical line from a point by using a plumb line.
</ul>
<h3>Step 1</h3>
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Block up the timber so that its flattest face (abcd) is level lengthwise.
<h3>Step 2</h3>
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Make a reference area in the middle of the face abcd at mnoq.
<h3>Step 3</h3>
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From X at the middle of ab and X' at the middle of cd, draw the line XX'.
<h3>Step 4</h3>
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Draw a perpendicular to XX' in the reference area.
<h3>Step 5</h3>
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Turn the timber 90 degrees so that face abcd is vertical and face adef is horizontal.
<h3>Step 6</h3>
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Make a reference area nqpr in the middle of face adef.
<h3>Step 7</h3>
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With Y at the middle of the line af and Y' at the middle of line de, draw the line YY'
<h3>Step 8</h3>
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Draw a perpendicular to YY' in the reference area that intersects point u. <i>[u is the intersection of the perpendicular drawn in face adef's reference area with the edge of the timber.]</i> This perpendicular intersects YY' at k'.
<h3>Step 9</h3>
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Make line k'u level across the width of the timber. <i>[The semi-circular object in the drawing is a plumb level.]</i>
<h3>Step 10</h3>
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<ul>
<li> Choose a point i on the end of the timber. The plumb level is placed on this point. The vertical distance iX is transferred [across the face] to vw.
<li> <i>[Draw Xw.]</i>
<li> The reference is the horizontal edge of the plumb level, oo'.
</ul>
<h3>Step 11</h3>
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<ul>
<li> At the other end of the timber, choose a point i1 and perform the same operation. The plumb level is placed on i1 and the vertical distance i1X is transferred to v1w1. The horizontal edge of the plumb level is still used as a reference.
<li> Draw line ww1.
</ul>
<h3>Step 12</h3>
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On the near face, put a plumb line against point Y. This defines point g on the line Xw and point t on the bottom face.
<h3>Step 13</h3>
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<ul>
<li> On the far face, put the plumb line against point Y' and mark point g' on the line X'w' and t' on the bottom face.
<li> Draw line tt' on the bottom face <i>[(after turning the timber so you can do this operation)]</i>.
</ul>
<h3>Step 14, the final result</h3>
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<ul>
<li> The four lines XX', YY', ww' and tt' represent the projections of the timber's axis onto the four faces.
<li> The axis passes through gg'.
</ul>
<h3>Conclusion</h3>
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The timber can now be aligned on the ground plan.
<p>
This procedure is directly applicable to assemblies in which the timbers' orientations are square to each other, like in roof trusses. In <a href="http://stereotomy-blog.blogspot.com/2012/03/le-piquage-or-theory-of-french-scribing.html">the previous post</a> in the series on the French scribe method, we skipped directly to the "good stuff" of laying out timbers with twisted orientations. The lining procedure is still useful in that case too. Our example was the layout of a hip rafter and king post. The king post will also be layed out with rafters, so it will be lined in the manner described here. That establishes a central axis, seen here:
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at the intersection of the green lines. The reference plane common to the hip rafter and king post will pass through that axis. Once we have the king post twisted to the correct orientation, we can use the plumb level to draw a level line through the central axis on both end faces of the timber. We then connect those to make the reference assembly lines that represent the reference plane.
<p>
Note: I found many translations of French carpentry terms at <a href="http://www.traditionaltimberframe.com/V1_0/index.php?mod=glossary&ac=glossary">Marc Guilhemjouan's pages</a> on traditional timber frame techniques.Tim Moorehttp://www.blogger.com/profile/09629429704217731021noreply@blogger.com2tag:blogger.com,1999:blog-1930479055984418545.post-30587727511454123932012-04-15T11:36:00.000+02:002012-04-15T11:36:38.231+02:00Stereotomy Blog - the Making OfFor those who might be interested in how I do the illustrations for Stereotomy Blog, I've started a series on my computer graphics blog <a href="http://shiny-dynamics.blogspot.com/">Shiny Dynamics</a>. I use the open-source 3D modeling program <a href="http://www.blender.org">Blender</a>; my first post explains how to draw a convincing wood grain in Blender. Future posts will cover topics such as:<br />
<ul><li> Non-realistic shading used by technical illustrators;<br />
<li> Emphasizing edges using the Freestyle rendering extension;<br />
<li> Integrating geometry and images from a 2D CAD program;<br />
<li> Drawing on objects using an external program, like in the images in the <a href="http://stereotomy-blog.blogspot.com/2012/03/le-piquage-or-theory-of-french-scribing.html">French scribe method post</a>.<br />
</ul>If your interests lie in that direction, check it out.Tim Moorehttp://www.blogger.com/profile/09629429704217731021noreply@blogger.com1tag:blogger.com,1999:blog-1930479055984418545.post-4854051876754961632012-03-23T11:46:00.000+01:002012-03-23T11:46:10.232+01:00A Real Roof ModelFaithful reader Rob Simpson made a cardboard model of the roof used in the <a href="http://stereotomy-blog.blogspot.fr/2011/09/devers-de-pas.html">devers de pas</a> series of posts:<br />
<br />
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Sometimes a physical model (and a nice glass of beer) are the best aids for understanding roof geometry!<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-n13zeiNgVF8/T2xT-K4QgBI/AAAAAAAAAdA/u4kXCmSgiyc/s1600/rob-rot.jpg" imageanchor="1" style=""><img border="0" height="320" width="240" src="http://3.bp.blogspot.com/-n13zeiNgVF8/T2xT-K4QgBI/AAAAAAAAAdA/u4kXCmSgiyc/s320/rob-rot.jpg" /></a></div>Tim Moorehttp://www.blogger.com/profile/09629429704217731021noreply@blogger.com2tag:blogger.com,1999:blog-1930479055984418545.post-75530532446519873612012-03-15T13:12:00.001+01:002012-03-27T17:58:37.348+02:00Le Piquage, or Theory of French ScribingWe are going to start with some informal observations about the geometry of intersecting timbers and, using those, explain <i>le piquage</i>, or "French scribing" as it is known in the English-speaking world. This layout technique is much older than the <i>rembarrement</i> we looked at <a href="http://stereotomy-blog.blogspot.com/2012/01/carpentry-model.html">last time</a>. There won't be any carpentry drawing today, but I find the technique fascinating, and we will soon tie it back to a drawing done at full-scale on the ground of the building site.<br />
<p>Here are the king post and a hip rafter from the model in the last entry:<br />
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The two timbers are not parallel, so we can take a long edge from each and displace them in space so that they intersect. Those intersecting lines define a plane. We can move the lines anywhere we want, so we can define an infinite family of planes that are all parallel to the edges of the timbers. Here are two such planes:<br />
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The plane on the right passes through the intersection of two roof surfaces on the hip rafter. We will use it as a reference to locate features on the two timbers. The left-hand plane is not very special; it is merely near the two timbers. But, its existence implies that any two timbers can be assembled against a plane. The planes and timbers can be tilted any way we like: for example, onto the ground, over a full-scale drawing of the layout:<br />
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(I've eliminated the carved top of the king post, which can be done later). This is a very useful fact for the carpenter. It means that any two timbers can be trial fitted on the ground, using a template, without needing to erect them into their final orientation in 3D space. The picture also presents the tantalizing prospect of being able to directly lay out the joint between the hip rafter and the king post, if only we could get them together into their final arrangement.<br />
<p>We can't do that before the joint is cut, because the timbers are solid. The closest we can do is move one of the timbers vertically, keeping the same orientation and longitudinal rotation. In other words, working backwards in time, we displace the hip rafter upward and fill in the joint:<br />
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On each timber we've drawn a line where our reference plane intersects them when they are assembled. If we choose a point on the hip rafter's reference line and move straight down to the level of the king post reference line, we will find a point on (or near) the king post that is coincident with the upper point when the timbers are joined together in their final arrangement. Also, we make a critical observation: each pairwise combination of faces in the hip rafter and king post have the same relative orientation in this configuration as they do when together, assuming everything is straight and flat. This implies that the intersection line between two displaced faces will have the same direction in space as they do in the final assembly, though the line will of course be in a different position. More formally, the direction of the intersection line is the vector cross product of the surface normals of the planes, and the normals are direction vectors without a fixed origin, so they are unaffected by the translation.<br />
<p>Let's see how we might lay out the joint, starting with the side of the hip rafter that is plumb i.e., the bottom in the final assembly. I am going to assume that the operations I describe can be performed accurately by a carpenter and save discussion of possible errors until the end. After the timbers are established in their correct relative orientation, the first step is to create a vertical reference line:<br />
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This is done with a special <a href="http://www.charpentiers.culture.fr/lesoutils/outilsdetracage/filsplomb">carpenter's plumb bob</a> with a hollow center so that it can be aligned with lines on the ground. Looking down from above, the plumb line is aligned with the intersection of the hip rafter face and protruding arris on the king post. The vertical plane defined by the plumb line and rafter face is projected and drawn on the king post with the use of dividers.<br />
<p>The horizontal distance from the plumb line projection on the king post to the plumb line is measured:<br />
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That distance is transferred and marked on the hip rafter.<br />
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That point marks the location where the king post arris intersects the rafter's face in the final layout.<br />
<p>Next, we find the direction of the intersection line of the two faces by laying a straight edge against both faces:<br />
<br />
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We draw a line on the hip rafter, but it is obviously not in the correct position. We know that the true intersection line must go through the king post arris, so we draw a line parallel to the direction line that goes through the point we marked previously:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-R_rdPTXTDPY/T2HWkvBuYtI/AAAAAAAAAZY/hG8wZtPSDME/s1600/0010.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://4.bp.blogspot.com/-R_rdPTXTDPY/T2HWkvBuYtI/AAAAAAAAAZY/hG8wZtPSDME/s400/0010.png" /></a></div><br />
We now have one side of the joint marked out.<br />
<p>The intersection line on the king post is coincident with the perpendicular line in this case, because the rafter surface is plumb. In order to find where the second side of the joint intersects the first, we measure the distance from the king post reference line up to the arris along the intersection line:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-rguikjuMmD0/T2HWlNvOW0I/AAAAAAAAAZk/QkfqWmEB_AQ/s1600/0011.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://2.bp.blogspot.com/-rguikjuMmD0/T2HWlNvOW0I/AAAAAAAAAZk/QkfqWmEB_AQ/s400/0011.png" /></a></div><br />
And mark it on the rafter:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-MXtak51fsWg/T2HXZy780RI/AAAAAAAAAZw/YwcI3YLLD1s/s1600/0012.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://3.bp.blogspot.com/-MXtak51fsWg/T2HXZy780RI/AAAAAAAAAZw/YwcI3YLLD1s/s400/0012.png" /></a></div><br />
That point is also where the intersection line crosses the plumb line reference. That is no great surprise because the plumb line touches the corresponding point on the king post, but this will not be true in general.<br />
<p>Now we establish the intersection line direction for the upper face of the king post:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-AeHST9pkGdU/T2HXaULXuKI/AAAAAAAAAZ8/VX_XK0TzP9I/s1600/0013.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://4.bp.blogspot.com/-AeHST9pkGdU/T2HXaULXuKI/AAAAAAAAAZ8/VX_XK0TzP9I/s400/0013.png" /></a></div><br />
And move it to cross the line from the lower plane at the correct point:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-CDZfOqiuECk/T2HXaomsSiI/AAAAAAAAAaI/n12ivsx8xO0/s1600/0014.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://2.bp.blogspot.com/-CDZfOqiuECk/T2HXaomsSiI/AAAAAAAAAaI/n12ivsx8xO0/s400/0014.png" /></a></div><br />
To summarize, I've redrawn the complete layout on this side without the construction lines:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-8a3205k_pys/T2HXbLjDq3I/AAAAAAAAAaU/4qGOjQ0pU8g/s1600/0015.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://3.bp.blogspot.com/-8a3205k_pys/T2HXbLjDq3I/AAAAAAAAAaU/4qGOjQ0pU8g/s400/0015.png" /></a></div><br />
Now we turn our attention to the other side of the hip rafter. Here the situation is slightly more complicated because there are two faces to lay out on the hip, and they aren't plumb. Nevertheless, the principles are the same. Start by establishing the plumb line reference against the arrises of the two rafters:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-4SRAt8XtjH4/T2HXbZCGpEI/AAAAAAAAAag/HVh8chPUDlQ/s1600/0016.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://3.bp.blogspot.com/-4SRAt8XtjH4/T2HXbZCGpEI/AAAAAAAAAag/HVh8chPUDlQ/s400/0016.png" /></a></div><br />
Drawing the projection of the plumb reference plane on the king post will be a bit more <i>artisanal</i> because there is no vertical plane to use as a reference.<br />
<p>The common reference point is measured and marked:<br />
<br />
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The direction of the intersection line between the two lower faces is found:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-1MRKyZ5kZkk/T2HYDUYEzfI/AAAAAAAAAa4/e6Ld8_6czhg/s1600/0018.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://1.bp.blogspot.com/-1MRKyZ5kZkk/T2HYDUYEzfI/AAAAAAAAAa4/e6Ld8_6czhg/s400/0018.png" /></a></div><br />
and transferred to the known intersection point:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-NeV2KAzs50M/T2HYDnx8RCI/AAAAAAAAAbE/UB5aqkWch3s/s1600/0019.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://2.bp.blogspot.com/-NeV2KAzs50M/T2HYDnx8RCI/AAAAAAAAAbE/UB5aqkWch3s/s400/0019.png" /></a></div><br />
Next we find the intersection line between the lower king post and upper rafter faces:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-IUiouWiT0Ro/T2HYEDZzPJI/AAAAAAAAAbQ/oQDdpdeP4yo/s1600/0020.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://3.bp.blogspot.com/-IUiouWiT0Ro/T2HYEDZzPJI/AAAAAAAAAbQ/oQDdpdeP4yo/s400/0020.png" /></a></div><br />
At the same time we find the distance along this line between the horizontal reference plane and the upper king post face. On the other side of the joint we were able to measure distance along the actual intersection line on the king post; here we cannot. However, we are measuring the distance on a line parallel to the real line, so the distance will be the same. Assuming everything the faces are flat, of course.<br />
<p>The intersection line is moved to its correct position on the upper hip rafter face, and the distance we just measured is marked:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-zCFX53FTKF8/T2HYEgMKPUI/AAAAAAAAAbc/LcBvzAv-veE/s1600/0021.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://4.bp.blogspot.com/-zCFX53FTKF8/T2HYEgMKPUI/AAAAAAAAAbc/LcBvzAv-veE/s400/0021.png" /></a></div><br />
The final intersection, between the two upper faces, is found:<br />
<br />
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And transferred:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-o2a4d__5SF8/T2HYnhT2wYI/AAAAAAAAAb0/faxevlXer2Y/s1600/0023.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://4.bp.blogspot.com/-o2a4d__5SF8/T2HYnhT2wYI/AAAAAAAAAb0/faxevlXer2Y/s400/0023.png" /></a></div><br />
<i>Voilà</i> the joint layout on this side:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-babwDOMceQU/T2HYoN8y6QI/AAAAAAAAAcA/cFxj_vx7PNw/s1600/0024.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://2.bp.blogspot.com/-babwDOMceQU/T2HYoN8y6QI/AAAAAAAAAcA/cFxj_vx7PNw/s400/0024.png" /></a></div><br />
To finish up we connect the layouts from the two sides across the top face:<br />
<br />
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and the bottom:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-DPtd9AkX2wY/T2HYpqN4OSI/AAAAAAAAAcU/2ZbASGqAycE/s1600/0026.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://2.bp.blogspot.com/-DPtd9AkX2wY/T2HYpqN4OSI/AAAAAAAAAcU/2ZbASGqAycE/s400/0026.png" /></a></div><br />
The connecting lines are parallel to the king post edges, which confirms that we did the layout correctly.<br />
<br />
This is the method that built the cathedral roofs of England and France in the Middle Ages. It is much more "concrete" than the <i>rembarrement</i> method we looked at last time: instead of dealing with abstract planes and using descriptive geometry to find their intersections, the actual timbers are directly used as a sort of graphical 3D calculator. The method relies on principles of geometry, but those do not have to be understood in order to apply it. On the other hand, quite a bit of skill would be required to carry out a real layout, and acquiring that skill was obviously a big part of the carpenter's apprenticeship.<br />
<p>This post is called "Theory of French Scribing" for a good reason. We have skipped many details of the process in order to concentrate on how the simple application of geometry leads to a layout method. The second case we encountered -- two timbers with no plumb faces -- is actually the most difficult to handle. It would be much more common to work with timbers with plumb faces, such as when laying out a principal rafter against the king post, or other parts of the frames. Here are some of the issues we blew off:<br />
<br />
<ul><li> We assumed that we can mark and measure horizontal and vertical lines in space by eye. This obviously takes a bit of practice. Today you can buy <a href="http://www.mehr-als-werkzeug.de/product/707269/Veritas-Log-Scribe-Scriber.htm">dividers with built in spirit levels</a>, intended for scribing the joints in a log cabin. Medieval carpenters didn't have those. On the other hand, small errors doesn't necessarily have much of an effect on our measurements. When we measured horizontally from the king post reference line out to the plumb line, we would have to be 8 degrees out of horizontal in order to have a 1 percent error. If an apprentice's dividers were that askew, the master would surely throw something at him!<br />
<li> We assumed that the timbers were straight and their faces flat. As I said the last time, one of the strengths of scribing is that it can handle warped timbers. An out-of-square timber is a special case of the general problem of laying out oblique faces that we just did, but in practice carpenters would use a faster method to deal with that. The intersection lines we found with the straight edge are usually close to the true intersections, so the results will be close, but judgment and skill are required.<br />
<li> Any errors we did make could be expensive. In practice, some margin would be left and trimmed away in the final trial fit of the two timbers.<br />
<li> Finally,this is personally theoretical because I've never done it! The pages of Mazerolle that explain the method are quite confusing. There is very little free information available on the Web on French scribing, and practically none explaining the method for oblique faces. I could be making a number of false assumptions.<br />
</ul>Nevertheless, we now know enough about scribing to be able to understand the notation on carpentry drawings that are intended for that layout method. We said nothing about how the timbers are positioned before scribing. We will cover that next time and see how to go from the drawing -- done at full scale -- to the arrangement and relative rotations of the timbers. I teased you by displaying a plan under the timbers, so that should give you a hint about positioning the timbers in their common reference plane.Tim Moorehttp://www.blogger.com/profile/09629429704217731021noreply@blogger.com4tag:blogger.com,1999:blog-1930479055984418545.post-72855082325299625212012-01-18T12:35:00.001+01:002012-01-19T08:56:23.810+01:00A Carpentry ModelWe're back with a new look, courtesy of Blender and the Freestyle renderer. I hope you like it. Personally, I'm satisfied with the "technical illustration" appearance of the 3D renderings , but I'm not so happy with the 2D drafting work flow and results and am considering moving away from Blender to a 2D CAD program for those.<br />
<br />
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We are going to look at a carpentry model that would have been used to teach roofing layout principles to apprentices and young carpenters. Today, <i>aspirants</i> and apprentices spend two hours a day in night school where study of these kinds of models still figures heavily. The model is not to scale and is missing common rafters, but it has features that appear in almost all roofs in the French tradition: principal and half rafters, hip rafters that are backed to meet the slope of their adjoining surfaces, and purlins that are supported by the principal and hip rafters. As we discussed in my last post, the surfaces of the different rafters lie on imaginary solids that are similar to the overal roof shape. This isn't always the case, but it is a guiding principle in the layout. This model even has moving parts, to illustrate the drafting construction for finding the hip rafter section:<br />
<br />
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<br />
We begin the plan by laying out the shape of the roof and the principal rafters:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-SLC6T-oU6IM/Txan-MrdMvI/AAAAAAAAAU4/bv176hGNfMQ/s1600/plan-0001.png" imageanchor="1" style=""><img border="0" height="283" width="400" src="http://2.bp.blogspot.com/-SLC6T-oU6IM/Txan-MrdMvI/AAAAAAAAAU4/bv176hGNfMQ/s400/plan-0001.png" /></a></div><br />
The "roof" is 64 x 102 centimeters, with a center height of 36.8 cm. The kingpost is 6.2cm x 6.8cm. Note that this shape is not similar to the rectangle of the whole roof. Therefore ridge lines don't meet the corners of the kingpost, giving its top a kind of funky truncated pyramid shape.<br />
<p>The principal rafters are 2.8cm wide. The top surface of the principal rafter lies 6.8 cm vertically below the roof surface. The bottom surface is 11.4745cm below the roof. There's no reason for that bizzare dimension, but I forget how I originally sized the rafter! In the drawing, the elevation view of the rafters is placed on top of a plan of half the roof. The cursive "N" stands for <i>niveau</i> or level. This indicates that that line is at the reference ground level of the whole plan.<br />
<p>Next, it is straightforward to construct the elevation view of the half rafters:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-H5jBQ8Whd5E/Txan_aslh2I/AAAAAAAAAVQ/P0oxDRZ3muA/s1600/plan-0004.png" imageanchor="1" style=""><img border="0" height="283" width="400" src="http://3.bp.blogspot.com/-H5jBQ8Whd5E/Txan_aslh2I/AAAAAAAAAVQ/P0oxDRZ3muA/s400/plan-0004.png" /></a></div><br />
The roof line and half rafter surfaces must intersect the principal surfaces at the center line of the king post. So, the intersections in the principal elevation view can be "swung down" with a compass to construct the other elevation; the half-rafter is then constructed parallel to the roof surface. The rafter footprints are directly constructed from the elevation views and the known width of the rafters. Here's what we have in 3d:<br />
<br />
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It is easy to see the odd shape of the top of the king post. Also, notice that the half rafter meets the king post at a slightly lower height than the principal rafters; again, this is due to the difference between the king post section and the shape of the whole roof.<br />
<p>The rafters are fixed to the king post with mortise and tenon joints. The layout of these joints is an interesting subproblem. the top of each tenon is horizontal so the rafter can be slid into place, and the bottom must follow the bottom surface of its rafter. The tenons coming in from the four sides of the king post need to meet nicely in the middle. <br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-C864-l8n37E/Txao-KtO9MI/AAAAAAAAAVo/4mnl4QKQghE/s1600/plan-0005.png" imageanchor="1" style=""><img border="0" height="283" width="400" src="http://3.bp.blogspot.com/-C864-l8n37E/Txao-KtO9MI/AAAAAAAAAVo/4mnl4QKQghE/s400/plan-0005.png" /></a></div><br />
I've sized the tenon for the half rafter at 0.8cm and given it a "V" shape that follows the ridge lines. The principal rafter tenons are sized proportionally to meet the half rafter tenons at the ridge lines. A portion of the principal rafter tenons is higher than the top of the half rafter tenons, so that is given a rectangular profile to avoid any voids in the middle of the king post:<br />
<br />
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An interesting result of constructing the tenons this way is that the interior of the mortise has the same shape as the roof:<br />
<br />
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We now turn our attention to the hip rafters. I'm going to give them the odd width of 5.037cm, for reasons that will soon become apparent. The hip is going to have a pentagonal shape: the top surfaces are aligned with the neighboring roof surfaces, the sides are plumb, and the bottom just touches the intersection of the upper surfaces of the principal and half rafters. The two plumb sides should have the same height, so in plan view the edges between the tops and sides intersect the ground plane on a line perpendicular to the axis of the hip. On the other hand, those edges should intersect the outer edges of the roof surface. How do we place a perpendicular line with the given constraints? The answer is a parallelogram construction:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-xbDve634m9Y/Txao-q6w68I/AAAAAAAAAV0/r5D8ss70TOQ/s1600/plan-0012.png" imageanchor="1" style=""><img border="0" height="283" width="400" src="http://3.bp.blogspot.com/-xbDve634m9Y/Txao-q6w68I/AAAAAAAAAV0/r5D8ss70TOQ/s400/plan-0012.png" /></a></div><br />
Two lines of the given length i.e., the desired width of the rafter, are extended from the corner vertex, perpendicular to the ridge line in plan. We extend a line parallel to the left roof edge from the end of the near line, and a parallel to the near roof edge from the end of the far line. These lines' intersections with the roof edge give us the line we are looking for: each of lines with which we started form a parallelogram that includes this new line, so that line must have the same length as the original lines [Edited per Chris' comments]. Lets draw in the rest of the hip:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-3J64TJdcvgc/Txao_KwdjhI/AAAAAAAAAV8/WWFQo4rwIWk/s1600/plan-0007.png" imageanchor="1" style=""><img border="0" height="283" width="400" src="http://2.bp.blogspot.com/-3J64TJdcvgc/Txao_KwdjhI/AAAAAAAAAV8/WWFQo4rwIWk/s400/plan-0007.png" /></a></div><br />
The inner edge of the hip rafter footprint is perpendicular to the axis of the rafter and meets the intersection of the upper rafter surfaces with the ground plane.<br />
<p>We can see the reason for the odd hip dimension now: the far edge of the hip meets the king post at the intersection of the left and far roof surfaces. We do this in order to give the hip enough width to have visible lip that sits against the near surface of the king post. Of course, this would give us problems if we constructed the far hip rafters, but we are not going to do that. This is not a scale model of a real roof; it only illustrates the layout on one side of a roof. We finish off the hip rafter's footprint by making its inner edge touch the "inner ridge line" formed by the top surfaces of the rafters.<br />
<p>It should be clear that I didn't start the layout of the hip by choosing a random dimension; I worked backwards from the king post and then measured the distance. However, the parallelogram construction for drawing the hip is a common technique, not just in French carpentry drawing but in other traditions as well, and it is important to be exposed to it.<br />
<p>After drawing the plan view of the hip rafter, we can derive the section of the rafter by using the technique in the <a href="http://stereotomy-blog.blogspot.com/2011/10/devers-de-pas-4.html">previous blog entry</a> for the triangular rafter. Draw an elevation view of the top of the hip rafter along its plan view, from point A to F, giving it the height of the king post at K. Choose a point X on the hip rafter and drop a line perpendicular to AK from there. This will be the point that is rotated down to the ground plane to begin the section construction, so we can choose the position of X to place Y, the rotated point, at a convenient location that is neither in the rafter footprint nor too far away in the middle of the drawing. Draw HZ, the intersection of the section plane with the ground. As explained in the last post, lines from Y to H and Z are the adjoining roof surfaces in the section drawing, so those give us the top of the rafter section.<br />
<br />
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The bottom of the section is found by locating the intersections of the bottom planes of the principal rafters with HZ. In the section view the lines showing these planes are parallel to the lines we already found for the roof surfaces. These new lines meet at U, giving us the bottom of the section.<br />
<p>When we turn the hip rafter on its hinge, we see that its cross section does line up with our construction lines, and the bottom of the rafter passes through point U:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-r2Ku3KjXrm4/Txap_nr4k3I/AAAAAAAAAXE/eugJhX2NeK0/s1600/0008.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://3.bp.blogspot.com/-r2Ku3KjXrm4/Txap_nr4k3I/AAAAAAAAAXE/eugJhX2NeK0/s400/0008.png" /></a></div><br />
The next task is to find the final shape of the hip rafter where it meets the king post and mark the required cuts on a drawing so that a carpenter could cut it out, either by transferring measurements onto the timber or by creating the drawing on the ground and laying the timber on top of it. This implies that we need to make drawings of the hip rafter in its true proportions i.e., with the length, height and width square to our view. The elevation we constructed for finding the hip rafter section is a good start: it shows true length, and we can easily add the height of the hip rafter to the drawing. We will add another view of the rafter from below to show the width.<br />
<p>From our plan and 3d views of the model, we know that the hip rafter wraps around the king post in a kind of ell called a <i>barbe</i> or beard. We can treat the near and right sides of the king post as planes that cut through the hip rafter:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-Em4YyDUvFx8/TxaqAd7qvrI/AAAAAAAAAXM/ifqbY7oR8Lg/s1600/0009.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://4.bp.blogspot.com/-Em4YyDUvFx8/TxaqAd7qvrI/AAAAAAAAAXM/ifqbY7oR8Lg/s400/0009.png" /></a></div><br />
The portion of the hip rafter that needs to be cut away lies on the "inside" of both planes, in a Boolean intersection of the half spaces defined by the planes.<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-bA2MsIFoX8o/TxaqAl4TMUI/AAAAAAAAAXY/jKFo-iZXwmU/s1600/0010.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://2.bp.blogspot.com/-bA2MsIFoX8o/TxaqAl4TMUI/AAAAAAAAAXY/jKFo-iZXwmU/s400/0010.png" /></a></div><br />
We will proceed by finding the intersections of those planes with the arrises of the hip rafter and then connecting them to form the lines of intersection of the hip rafter and king post surfaces. When these lines are marked out on the real timber, the carpenter can saw along them to establish the needed cuts. Obviously he shouldn't saw all the way through the wood, but it is reasonably straight-forward to saw along both sets of lines until their intersection, and no further.<br />
<p>In the plan view, extend the line of the near (red) king post surface to intersect all the edges of the hip rafter:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-uBOIxLsIvzw/TxapAb2qDUI/AAAAAAAAAWY/oksoaIAD5r0/s1600/plan-0010.png" imageanchor="1" style=""><img border="0" height="283" width="400" src="http://4.bp.blogspot.com/-uBOIxLsIvzw/TxapAb2qDUI/AAAAAAAAAWY/oksoaIAD5r0/s400/plan-0010.png" /></a></div><br />
The left and right lines of the rafter in plan view represent upper and lower edges of the vertical surfaces of the rafter, so we will need to plot the intersections of each of them with the king post surfaces. There is only one top arris on the rafter, of course. We bring the intersections from the plan view up into the elevation view of the hip rafter, and from there into the rafter's bottom view. (Dashed lines in the rafter views show lines that are hidden by surfaces that are closer to the viewer; for various reasons the dashes may not show up well. I am hoping to fix that in future articles).<br />
<p>The procedure for the right (blue) king post plane is identical:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-cdWfvYTqs-M/Txaqs-xdsJI/AAAAAAAAAXk/MOLOuxe_hEI/s1600/plan-0011.png" imageanchor="1" style=""><img border="0" height="283" width="400" src="http://1.bp.blogspot.com/-cdWfvYTqs-M/Txaqs-xdsJI/AAAAAAAAAXk/MOLOuxe_hEI/s400/plan-0011.png" /></a></div><br />
The joint between the hip rafter and the king post has been completely laid out. If the drawing were done at full scale on the workshop floor or construction site, the timber could be placed on top of e.g., the elevation drawing in order to transfer the intersections in the drawing to the timber. Marks on the right side of the timber can be transferred directly; the other marks need to be brought up the sides of the timber using a square.<br />
<p>Both the drafting and the timber marking methods are called <i>rembarrement</i>. This has nothing to do with rude behavior. The word refers to "pushing back" a point along a line from one context to another. When drafting the hip rafter, we moved the intersection points from plan view, where they were easy to find, into the much more useful elevation view. The carpenter moves marks from the floor up to edges using a square or plumb bob, perhaps making marks on intervening edges. These methods are still in use in carpentry drafting and layout. <i>Rembarrement</i> works well when using the straight and square timber that is easily available today. If we placed the hip rafter timber over its full-scale drawing and found that it was warped or out of square, we could probably fake it: use a corner and edge of the timber as a reference and mark out the lines. However, if the king post timber was out of square we would be kind of stuck. We don't use the real king post to lay out the hip rafter, but an abstraction of its surface planes that are perfectly flat and square. We could proceed by "fixing" those parts of the king post that do intersect with the rafter to be square. As I understand it, this is the "square rule" practice that started appearing in America in the early 18th century. <br />
<p>In Europe, carpenters had been dealing with out-of-square timbers in their carpentry drawings since the 13th century (at least). In the next installment we'll look at the layout method they used and how it is incorporated in the carpentry drawing.Tim Moorehttp://www.blogger.com/profile/09629429704217731021noreply@blogger.com8tag:blogger.com,1999:blog-1930479055984418545.post-54186345039226360222011-10-17T13:14:00.002+02:002011-10-17T13:14:26.143+02:00devers de pas (4)In the last post we looked at a "folding" method of developing the ground plane footprints -- <i>devers de pas</i> -- for a timber that works with the actual cross section of that timber. Now we will apply that to several complex shapes for the last rafters in our little model. First though, a correction...<br />
<h3>Unwrapping the Onion</h3>In the comments to the <a href="http://stereotomy-blog.blogspot.com/2011/10/in-this-post-we-will-add-only-one.html">part 2</a>, Chris Hall pointed out a problem with the inner surfaces of the near and far roof slabs: they do not intersect the left and right slabs at the ridge lines (in plan view). I gave the rafters at H and G the same width as the principal rafters at E and F. This does not affect our exercises for finding the footprints for these rafters, but it messes up another aspect of the layout. Consider the solid made by the inner surfaces of the rafters. If this is congruent to the shape of the main roof, then not only do many aspects of the layout become much neater, but ideals of symmetry (and, I suspect, various spiritual ideals as well) are satisfied. In a real roof there are several such solids formed by the inner and outer surfaces of the common rafters, purlins, and principal rafters, and they all should be congruent. The desired configuration in our model looks like this:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-5zsqiSH05nM/TpwLS7pJ0FI/AAAAAAAAAOg/bINEaUKt0j0/s1600/0035.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://3.bp.blogspot.com/-5zsqiSH05nM/TpwLS7pJ0FI/AAAAAAAAAOg/bINEaUKt0j0/s400/0035.png" /></a></div>Notice the similar shapes of the inner and outer surfaces, with all the rafters lying between them. In solid geometry terms, the inner surface is the same as the outer, scaled uniformly about a point in the ground plane lying directly below the peak of the roof. The inner solid produced by my original layout, which I'm not illustrating here, would be an oddly skewed version of the main roof surface.<br />
<p>How do we layout the correct inner surface of the plan? We said that the inner surface has been scaled around a point underneath the peak, so the inner peak must also be directly under the outer peak. Therefore, in each elevation view that cuts through the two peaks, the distance between the two will be the same. The dimensions and layout of the principal rafters are given, so the widths of the rafters in the others surfaces will be determined by this principle. Here's a corrected version of the plan:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-QLezJ4YsQQg/TpwKrJJG9rI/AAAAAAAAANA/KPy59gwrLJ8/s1600/dp2plan-crop0035.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://1.bp.blogspot.com/-QLezJ4YsQQg/TpwKrJJG9rI/AAAAAAAAANA/KPy59gwrLJ8/s400/dp2plan-crop0035.png" /></a></div>An interesting result is that the distance between the inner and outer surfaces will be different for each different slope in the roof, and so the rafters will all have different dimensions. This sounds like a lot of work for the carpenters, but I suppose it doesn't matter if you have to resaw everything anyway.<br />
<p><h3>Back to Footprints</h3>With that out of the way, we return to laying out rafters. The next rafter will have a completely irregular cross section with one side lying against the near surface of the roof. We start as we did for the rafter at D by drawing an elevation view of the rafter and folding a perpendicular cross section plane down to the ground:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-n-TEk3Qov3U/TpwJ5CSjjmI/AAAAAAAAALI/D6AeEvAGxGY/s1600/dp2plan-crop0026.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://2.bp.blogspot.com/-n-TEk3Qov3U/TpwJ5CSjjmI/AAAAAAAAALI/D6AeEvAGxGY/s400/dp2plan-crop0026.png" /></a></div>and then find,in the cross section, the edge that lies against the near surface. This is the same construction we did last time, running a line from point 2 to the intersection of the folding line and the gutter line:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-ozNrlg0Gqt0/TpwJ5WCxfRI/AAAAAAAAALQ/07v8DWNnqM0/s1600/dp2plan-crop0027.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://1.bp.blogspot.com/-ozNrlg0Gqt0/TpwJ5WCxfRI/AAAAAAAAALQ/07v8DWNnqM0/s400/dp2plan-crop0027.png" /></a></div>Next we draw the rest of the cross section:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-pvv03_qJ25U/TpwJ5p6bUmI/AAAAAAAAALk/Bxhkvwl49Do/s1600/dp2plan-crop0028.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://4.bp.blogspot.com/-pvv03_qJ25U/TpwJ5p6bUmI/AAAAAAAAALk/Bxhkvwl49Do/s400/dp2plan-crop0028.png" /></a></div><br />
At this point in 3d, we can place the rafter in the model and see what we have:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-D9KM6HhVSc0/TpwLEZJ7_MI/AAAAAAAAANk/iYezju3xshA/s1600/0028.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://3.bp.blogspot.com/-D9KM6HhVSc0/TpwLEZJ7_MI/AAAAAAAAANk/iYezju3xshA/s400/0028.png" /></a></div>Next, we find where the points 4 and 6 end up in the footprint:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-Ic5PcCV_oLo/TpwJ6BE7vuI/AAAAAAAAALs/EECaAhzMe3U/s1600/dp2plan-crop0029.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://2.bp.blogspot.com/-Ic5PcCV_oLo/TpwJ6BE7vuI/AAAAAAAAALs/EECaAhzMe3U/s400/dp2plan-crop0029.png" /></a></div>Point 9 is found by extending a parallel to the elevation cross section -- the ridge line -- through point 6 to the edge of the roof. To find point 11, we do the same construction to transfer a surface intersection from the cross section view to the plan, only in reverse. The line between 2 and 4 is extended to intersect the fold line at point 8; the line from point A to 8 is then the left edge (and <i>dévers de pas</i> line) of the footprint. Point 11 is located on it with the same parallel method used to find point 9.<br />
<p>To find the last vertex of the footprint, we transfer the last two edges:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-mVGTskLIBdo/TpwJ6cjKL1I/AAAAAAAAAL0/ME7qMY5MDKs/s1600/dp2plan-crop0030.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://2.bp.blogspot.com/-mVGTskLIBdo/TpwJ6cjKL1I/AAAAAAAAAL0/ME7qMY5MDKs/s400/dp2plan-crop0030.png" /></a></div>The intersection of the edge between points 4 and 5 is outside the crop line of the plan, but I assure you that the construction is still correct :). The edges of the footprint extended from points 11 and 9 intersect to give us point 12.<br />
<p>In 3D, we see how the lines in the cross section come down to give us the edges of the footprint:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-PSivYhcrZ-c/TpwLEjHRzeI/AAAAAAAAANs/-VaDwYGAKmI/s1600/0030.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://2.bp.blogspot.com/-PSivYhcrZ-c/TpwLEjHRzeI/AAAAAAAAANs/-VaDwYGAKmI/s400/0030.png" /></a></div>In passing, Mazerolle describes a bit of geometrical trickery that would allow us to find point 2 and the initial edge without drawing an elevation of the rafter. We drew the elevation view of the near roof surface when we constructed the rafter at point H. We will use it again to draw an arc centered at K and tangent to the outer roof surface:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-g6NOr-dptiU/TpwKRSubO1I/AAAAAAAAAME/-s6IB83v0OU/s1600/dp2plan-crop0031.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://3.bp.blogspot.com/-g6NOr-dptiU/TpwKRSubO1I/AAAAAAAAAME/-s6IB83v0OU/s400/dp2plan-crop0031.png" /></a></div>Now, the plane of the elevation view at H passes through point K (on the ground) and is perpendicular to the roof surface. The cross section plane of rafter A also contains K. It is perpendicular to the roof surface because by definition it is perpendicular to the upper edge of the rafter, which lies in that roof surface. So, the rafter cross section plane must intersect the roof elevation at the normal line from K to the roof surface:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-RHHOBnwdogs/TpwLEzn-S2I/AAAAAAAAAN4/_0te46s1QU8/s1600/0031.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://4.bp.blogspot.com/-RHHOBnwdogs/TpwLEzn-S2I/AAAAAAAAAN4/_0te46s1QU8/s400/0031.png" /></a></div>Therefore, the distance from K to the roof plane will be the same in the two planes. We run a line from the intersection of the fold line and roof gutter tangent to the arc we drew back to the ridge line, which gives use point 2 and the first edge.<br />
<p>I don't see that this is easier than drawing the elevation, but the geometry that supports this construction is interesting.<br />
<p> The next rafter will have an equilateral triangle as a cross section. Mazerolle says that "The rafter B ... is obtained in the same manner as that which came before." That's not exactly true in the case of the development in the book, and we'll also add an additional twist that forces us to proceed differently: the bottom of the rafter will be level. We do start by constructing an elevation view of the rafter at B:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-ouYdPdIQYIA/TpwKRlf7aKI/AAAAAAAAAMM/RdEW54jYfmI/s1600/dp2plan-crop0032.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://4.bp.blogspot.com/-ouYdPdIQYIA/TpwKRlf7aKI/AAAAAAAAAMM/RdEW54jYfmI/s400/dp2plan-crop0032.png" /></a></div>If we ran the fold line for our cross section view through the center point K, that would put the view rather far from point B, where we will draw the footprint. In fact, we don't need to draw the fold line through K; any perpendicular to the ridge line will do. So, we've chosen a fold line that is closer to B and get our initial point.<br />
<p>We will give the triangle a size such that the footprint of the rafter will just touch the intersection of the inner surfaces of the left and far roofs. This implies that the rafter will "run up" the edge formed by those surfaces; therefore, in the cross section view, the lower edge of the triangle will also touch that intersection. We know how to find those surfaces and intersections in the cross section view:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-_Xzq5e1WIHs/TpwKRxqxVDI/AAAAAAAAAMc/4HfW9qIvxZI/s1600/dp2plan-crop0033.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://3.bp.blogspot.com/-_Xzq5e1WIHs/TpwKRxqxVDI/AAAAAAAAAMc/4HfW9qIvxZI/s400/dp2plan-crop0033.png" /></a></div>We find the intersection of the fold line and the outer roof surfaces and connect them to the starting point. Then, the inner surface lines in the cross section come from the intersection of the fold line and inner ground lines which must be parallel to the outer surface lines. Their crossing at the ridge line gives us the position of the lower edge in the cross section. The cross section looks good in 3D:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/--EuGwrPelW0/TpwLFEvIsSI/AAAAAAAAAOI/3ewE11uJePI/s1600/0033.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://3.bp.blogspot.com/--EuGwrPelW0/TpwLFEvIsSI/AAAAAAAAAOI/3ewE11uJePI/s400/0033.png" /></a></div>The footprint is established with the same methods used before: <br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-SBv9ojw2qJs/TpwKScCDwrI/AAAAAAAAAMk/Ij-RS6HGfxo/s1600/dp2plan-crop0034.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://3.bp.blogspot.com/-SBv9ojw2qJs/TpwKScCDwrI/AAAAAAAAAMk/Ij-RS6HGfxo/s400/dp2plan-crop0034.png" /></a></div>The two outer edges pass through the intersections of those edges in the cross section and the fold line. The end points of the third edge are found with parallels through the cross section view. As expected, the footprint just touches the intersection of the inner ground lines. The footprint agrees with the 3D view of the rafter:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-ZRmTy-02wYY/TpwLFc3x91I/AAAAAAAAAOQ/uEKgWL-xp6E/s1600/0034.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://1.bp.blogspot.com/-ZRmTy-02wYY/TpwLFc3x91I/AAAAAAAAAOQ/uEKgWL-xp6E/s400/0034.png" /></a></div>The last rafter, erected at C, is a hexagon that lies against the far roof surface. The cross section and footprint are constructed using the same methods we've used already:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-_cxyh0yqNPk/TpwKreRBuzI/AAAAAAAAANI/2z0pfhSc2II/s1600/dp2plan-crop0036.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://1.bp.blogspot.com/-_cxyh0yqNPk/TpwKreRBuzI/AAAAAAAAANI/2z0pfhSc2II/s400/dp2plan-crop0036.png" /></a></div>As we've come to expect, the 3D view shows the rafter sitting nicely on its footprint:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-UIe9ZpkcCd8/TpwLTALWOUI/AAAAAAAAAOs/zDuQ6Uc_iUc/s1600/0036.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://1.bp.blogspot.com/-UIe9ZpkcCd8/TpwLTALWOUI/AAAAAAAAAOs/zDuQ6Uc_iUc/s400/0036.png" /></a></div>As a justification for fooling with such an exotic footprint, Mazerolle makes the point that if the rafter cross section was circular, then the inscribed hexagon could be used to find the elliptical footprint, with the help of a <i>pistolet</i> or French curve:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-Q0oiHQQyYpg/TpwKrkCUo_I/AAAAAAAAANY/Tr7gGk9jstE/s1600/dp2plan-crop0037.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://1.bp.blogspot.com/-Q0oiHQQyYpg/TpwKrkCUo_I/AAAAAAAAANY/Tr7gGk9jstE/s400/dp2plan-crop0037.png" /></a></div>The draftsman would run the French curve through points of the footprint to find the shape that is known to be an ellipse. I don't have a set of French curves handy, and they would be awkward to use with Blender, so I made the ellipse by rotating and scaling a circle. That was tricky, because none of the lines between vertices of the hexagon lie on the major or minor axes of the ellipse. We could probably change the orientation of the hexagon a bit to make this method more practical.<br />
<p>We are now equipped to find the <i>devers de pas</i> footprints, and the corresponding <i>dévers de pas</i> surface lines, in just about any situation. While Mazerolle's carpentry drawings usually use the first technique from <a href="http://stereotomy-blog.blogspot.com/2011/10/in-this-post-we-will-add-only-one.html">part 2</a>, where we used a <i>trait carré</i> normal line from an existing elevation view, we should now be able to handle whatever he throws at us in the world of <i>devers de pas</i>.Tim Moorehttp://www.blogger.com/profile/09629429704217731021noreply@blogger.com5tag:blogger.com,1999:blog-1930479055984418545.post-32448796665242959582011-10-05T14:05:00.002+02:002011-10-17T13:15:28.822+02:00devers de pas (3)In this post we will add only one rafter to our evolving model, but we will show two different ways to determine its footprint -- or <i>devers de pas</i> -- on the plan. We are going to add a hip rafter at the intersection of the right and near roofs. We will keep this rafter square, so we must choose which roof to align it with, if any; we will orient it so that its top face aligns with the right roof surface.<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-4gwcVZbhlZo/ToxDvmnOhjI/AAAAAAAAAJw/vXJByv18KDY/s1600/0013.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://3.bp.blogspot.com/-4gwcVZbhlZo/ToxDvmnOhjI/AAAAAAAAAJw/vXJByv18KDY/s400/0013.png" /></a></div>In the plan, we show the roof ridge lines. One arris of the rafter will run along the ridge line from D at ground level to K at the roof peak. We will set the width along the gutter line of the rafter, rather than sizing the rafter itself. In my model I've set that width at 1.5 cm and marked the end of the edge along the gutter as I.<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-m5mB3I8LJ6E/ToxCan9fMoI/AAAAAAAAAIQ/eQrBjA0CXnY/s1600/dp2plan-crop0013.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://4.bp.blogspot.com/-m5mB3I8LJ6E/ToxCan9fMoI/AAAAAAAAAIQ/eQrBjA0CXnY/s400/dp2plan-crop0013.png" /></a></div>We proceed exactly as we did with rafter G in the <a href="http://stereotomy-blog.blogspot.com/2011/09/devers-de-pas-2.html">last post</a>. In fact, we can do even less work, because we don't need to construct an elevation view of this roof surface: we already have the elevation view at the top of the plan. We will use it to construct the surface normal by dropping a perpendicular -- <i>trait carré</i> -- from the top of the roof line in the elevation view. We draw the intersection at point R.<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-8BjFOEXJOy0/ToxCa_jEyNI/AAAAAAAAAIY/0K7JSlpj3-c/s1600/dp2plan-crop0014.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://2.bp.blogspot.com/-8BjFOEXJOy0/ToxCa_jEyNI/AAAAAAAAAIY/0K7JSlpj3-c/s400/dp2plan-crop0014.png" /></a></div>In this roof construction the elevation view is only valid for a cross section taken through the point K. For simple A-frame style roofs and many more complex shapes this principal elevation view might be useful all along the roof, but the difficulty is the same: we can't use R directly, because the elevation view is placed arbitrarily high in the drawing. It is aligned with the center axis of the plan, so can drop a vertical line from R to the point R', which lies on the line where the elevation view was made, to find where the roof surface normal intersects the plan.<br />
<p>A 3D view:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-InG749OHfp8/ToxDvyZ2g6I/AAAAAAAAAJ4/8_4VNfbfPSQ/s1600/0014.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://4.bp.blogspot.com/-InG749OHfp8/ToxDvyZ2g6I/AAAAAAAAAJ4/8_4VNfbfPSQ/s400/0014.png" /></a></div>We know that we can connect D with R' to get the <i>dévers de pas</i> line at the intersection of the near surface plane of the rafter with the plan:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-jEq_ZJXJo2c/ToxCbHMhQvI/AAAAAAAAAIg/WcKihLQW3jk/s1600/dp2plan-crop0015.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://4.bp.blogspot.com/-jEq_ZJXJo2c/ToxCbHMhQvI/AAAAAAAAAIg/WcKihLQW3jk/s400/dp2plan-crop0015.png" /></a></div>and the 3D view of that plane shows that we did this correctly.<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-xNoWcYcODh0/ToxDv3gR1OI/AAAAAAAAAKA/oxjhsbtpMyo/s1600/0015.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://3.bp.blogspot.com/-xNoWcYcODh0/ToxDv3gR1OI/AAAAAAAAAKA/oxjhsbtpMyo/s400/0015.png" /></a></div>We finish the footprint on the plan by drawing a line parallel to R'D through point I. The inner (left) face of the rafter intersects the ground on the line determined in the elevation view for all rafters in this roof.<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-ZhNFtKCTacg/ToxCbcr6MVI/AAAAAAAAAIo/Qx2FU0MedWE/s1600/dp2plan-crop0016.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://2.bp.blogspot.com/-ZhNFtKCTacg/ToxCbcr6MVI/AAAAAAAAAIo/Qx2FU0MedWE/s400/dp2plan-crop0016.png" /></a></div>We will now draw the rafter footprint by another method which, while it requires more work, is also more versatile. We will construct the cross section of the rafter, project those edges onto the plan, and then connect them up to find the footprint. The first step is to construct an elevation view of the rafter:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-pMGVlmQBK4I/ToxCbswoFOI/AAAAAAAAAIw/KMyTglsNdVo/s1600/dp2plan-crop0017.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://1.bp.blogspot.com/-pMGVlmQBK4I/ToxCbswoFOI/AAAAAAAAAIw/KMyTglsNdVo/s400/dp2plan-crop0017.png" /></a></div>Using the height h of the roof, we extend a perpendicular line from KD to point S. DS is a view of the arris of the rafter that runs along the ridge line. Next, draw a perpendicular line to DS running through the center point K, which intersects DS at T. Drop an arc KT down to KD, which represents the ground in this view. The arc intersects KD at point U.<br />
<p>We are laying out a view of the cross section of the rafter. It will be most convenient to draw the cross section on top of the rest of the plan, even though will need to be careful to keep these two views straight. In 3D we can see that we are taking the cross-section plane of the rafter that passes through the center point of the ground plan:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-uNOJkfXIrEs/ToxDwC0BjFI/AAAAAAAAAKI/cnHSD2uF5CQ/s1600/0017.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://4.bp.blogspot.com/-uNOJkfXIrEs/ToxDwC0BjFI/AAAAAAAAAKI/cnHSD2uF5CQ/s400/0017.png" /></a></div>and folding it along the line KS down to the plan:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-tdL2ZlapyoU/ToxDwfUw_4I/AAAAAAAAAKQ/4kZDWqCi3eI/s1600/0018.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://4.bp.blogspot.com/-tdL2ZlapyoU/ToxDwfUw_4I/AAAAAAAAAKQ/4kZDWqCi3eI/s400/0018.png" /></a></div>U was on the top edge of the rafter, and now we are working with it on top of the plan. How does the view of the cross section plane and the rest of the plan relate? We obviously can't carry points directly from one view to the other, but we can make two helpful observations. First, any point on the line KS will be the same in both views, because that was the axis of rotation of our fold. Second, distances measured perpendicular to KU will be the same in both views because KU is perpendicular to the folding rotation. This is equivalent to stating that parallels to KU are the same in both views.<br />
<p>The next task is constructing the upper edge of the rafter on the cross section. This is the major constraint in the layout of the cross section because the rafter is specified to lie against the right roof surface. This edge is the intersection of the cross section plane and the roof surface; in order to draw it we need to find two points on that intersection and connect them. The first point is U. The second is the intersection of the fold line KS with the gutter line of the roof surface. They intersect at V:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-qMLoJBN0Jc0/ToxC873jZ3I/AAAAAAAAAI4/fqI88Dbtv-s/s1600/dp2plan-crop0019.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://1.bp.blogspot.com/-qMLoJBN0Jc0/ToxC873jZ3I/AAAAAAAAAI4/fqI88Dbtv-s/s400/dp2plan-crop0019.png" /></a></div>in 3D:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-Bt9N_DjrLA8/ToxEHAhVjUI/AAAAAAAAAKY/i42b1RyYP2s/s1600/0019.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://4.bp.blogspot.com/-Bt9N_DjrLA8/ToxEHAhVjUI/AAAAAAAAAKY/i42b1RyYP2s/s400/0019.png" /></a></div>The lower rafter surface also intersects the cross section plane at a point on KS, which we mark as V'. The intersection line most be parallel to line for the upper surface, KS, so we draw that passing though V':<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-YP8J-cwXUCQ/ToxC87ERqgI/AAAAAAAAAJA/lv_-zF1V6HI/s1600/dp2plan-crop0020.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://3.bp.blogspot.com/-YP8J-cwXUCQ/ToxC87ERqgI/AAAAAAAAAJA/lv_-zF1V6HI/s400/dp2plan-crop0020.png" /></a></div><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-3D38zUYQwE8/ToxEHaLSvfI/AAAAAAAAAKg/cEEbNFwymGI/s1600/0020.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://2.bp.blogspot.com/-3D38zUYQwE8/ToxEHaLSvfI/AAAAAAAAAKg/cEEbNFwymGI/s400/0020.png" /></a></div>The cross section is rectangular, so make a perpendicular line to UV at U. This line intersects the inner surface intersection line at X, giving us another point on the cross section:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-FgvKsG5Gdh0/ToxC9LXO4VI/AAAAAAAAAJI/1s2p2VP_-Ck/s1600/dp2plan-crop0021.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://1.bp.blogspot.com/-FgvKsG5Gdh0/ToxC9LXO4VI/AAAAAAAAAJI/1s2p2VP_-Ck/s400/dp2plan-crop0021.png" /></a></div>It intersects KS at y, which is of course in the plane of the ground plan:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-jP-cVcthH6k/ToxEHnVszFI/AAAAAAAAAKo/DyxKnyZ0TYk/s1600/0021.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://1.bp.blogspot.com/-jP-cVcthH6k/ToxEHnVszFI/AAAAAAAAAKo/DyxKnyZ0TYk/s400/0021.png" /></a></div>Now, using the fact that parallels to KU are the same in both views, we extend a line from I to intersect UV at point a:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-YVvpPz1O150/ToxC9egCUaI/AAAAAAAAAJQ/LGlcwOmAL0U/s1600/dp2plan-crop0022.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://2.bp.blogspot.com/-YVvpPz1O150/ToxC9egCUaI/AAAAAAAAAJQ/LGlcwOmAL0U/s400/dp2plan-crop0022.png" /></a></div>This gives us a 3rd point on the cross section. Drawing a parallel to UX through a gives us point z and the complete intersection.<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-N5EEwGEDh4E/ToxEHqcAu8I/AAAAAAAAAKw/FyvOLX_tqP0/s1600/0022.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://1.bp.blogspot.com/-N5EEwGEDh4E/ToxEHqcAu8I/AAAAAAAAAKw/FyvOLX_tqP0/s400/0022.png" /></a></div>As one would hope, this cross section aligns perfectly with the actual cross section of the rafter:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-tCub0tGOUSs/ToxEH9WpZtI/AAAAAAAAAK4/6HcRovYm3uY/s1600/0023.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://3.bp.blogspot.com/-tCub0tGOUSs/ToxEH9WpZtI/AAAAAAAAAK4/6HcRovYm3uY/s400/0023.png" /></a></div>Back to <i>devers de pas</i>. Point y lies in the plane of the near surface of the rafter and in the ground plan, as does D. So, we connect them to get the <i>dévers de pas</i> line Dy at the intersection of the near rafter plane and the ground:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-yy-ZsfRHP_I/ToxDNJVSFUI/AAAAAAAAAJg/Odwg3JdAR7Q/s1600/dp2plan-crop0024.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://1.bp.blogspot.com/-yy-ZsfRHP_I/ToxDNJVSFUI/AAAAAAAAAJg/Odwg3JdAR7Q/s400/dp2plan-crop0024.png" /></a></div>In fact, Dy does run along the near edge of the rafter footprint:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-nCN9zHrD7XQ/ToxEU6SibSI/AAAAAAAAALA/RJV6uTcZJdg/s1600/0024.png" imageanchor="1" style=""><img border="0" height="225" width="400" src="http://4.bp.blogspot.com/-nCN9zHrD7XQ/ToxEU6SibSI/AAAAAAAAALA/RJV6uTcZJdg/s400/0024.png" /></a></div>We finish the footprint on the plan by using a parallel to Dy running through point I, and it agrees perfectly with the footprint we constructed by dropping a <i>trait carré</i> from the rafter elevation.<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-XvZto3iO8XY/ToxDNaXrQuI/AAAAAAAAAJo/62xYkVFux5o/s1600/dp2plan-crop0025.png" imageanchor="1" style=""><img border="0" height="400" width="300" src="http://3.bp.blogspot.com/-XvZto3iO8XY/ToxDNaXrQuI/AAAAAAAAAJo/62xYkVFux5o/s400/dp2plan-crop0025.png" /></a></div><p>This folding method works in situations where we can't conveniently find a surface normal for a face of a rafter. Chris Hall used a variation of the technique in his <a href="http://thecarpentryway.blogspot.com/2010/11/x-marks-spot.html">X Marks the Spot</a> series of posts. It was useful there because the orientation of the timbers was arbitrary and not related to any roof surface. It also works in situations where the rafter cross section is not square, or even polygonal (!), as we shall see in the next post. Stay tuned.Tim Moorehttp://www.blogger.com/profile/09629429704217731021noreply@blogger.com16tag:blogger.com,1999:blog-1930479055984418545.post-61089879377838021102011-09-29T00:04:00.002+02:002011-10-17T13:16:14.873+02:00devers de pas (2)In the first post in this series, I introduced the "devers de pas" method and gave an example of its use. Now we will see how to construct the footprint at ground level in a few different situations. In future posts I will cover more complicated cases.<p>Our plan is the same as we used in the first post.<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/-YYa8YZjlJLY/ToNW2yc9J7I/AAAAAAAAAFw/94dcwv_YByk/s1600/dp2plan-crop0001.png"><img style="cursor:pointer; cursor:hand;width: 300px; height: 400px;" src="http://2.bp.blogspot.com/-YYa8YZjlJLY/ToNW2yc9J7I/AAAAAAAAAFw/94dcwv_YByk/s400/dp2plan-crop0001.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657461056163620786" /></a><br />
This shows a trapezoidal ground plan and a cross section of the roof at the principal rafters, which are placed points E and F. The result will be a pyramid with the faces meeting at the highest point, K. We draw in the ridge lines where each face meets its neighbor:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/-z5mKvLcor6Q/ToNW21NWppI/AAAAAAAAAF4/5eyc4lPb_qY/s1600/dp2plan-crop0002.png"><img style="cursor:pointer; cursor:hand;width: 300px; height: 400px;" src="http://3.bp.blogspot.com/-z5mKvLcor6Q/ToNW21NWppI/AAAAAAAAAF4/5eyc4lPb_qY/s400/dp2plan-crop0002.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657461056903489170" /></a><br />
and take a look at the situation in 3D:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/-O2WXUEGSbrI/ToNZFuxWvXI/AAAAAAAAAGY/i6wXXuwY-AQ/s1600/0002.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://2.bp.blogspot.com/-O2WXUEGSbrI/ToNZFuxWvXI/AAAAAAAAAGY/i6wXXuwY-AQ/s400/0002.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657463511896735090" /></a><br />
The left and right (looking from the top) roof faces have the same slope, but the top and bottom faces are at different slopes.<br />
<p>We are going to run rafters along all the faces and all the ridge lines, with different cross sections too. We will assume that each rafter goes all the way to the peak of the roof, even though that would produce an impossible traffic jam. Once we've found a timber's footprint, we will show it in 3D with the top sawed off. We will first find the footprint of the principal rafters:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/-4AiRj_hceQE/ToNZGB4nUZI/AAAAAAAAAGg/W4o1mPP9i9A/s1600/0003.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://1.bp.blogspot.com/-4AiRj_hceQE/ToNZGB4nUZI/AAAAAAAAAGg/W4o1mPP9i9A/s400/0003.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657463517027455378" /></a><br />
This should be pretty easy. The top and bottom edges of the footprints are already shown on the plan: those are the vertical faces of the rafters. In order to find the inner edge of the footprint, we need to look at the elevation drawing. We find the point where the inner edge of the rafter intersects the ground plane, and run it down into the<br />
plan:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/-V7ZN1L9BkfU/ToNW3GOwDGI/AAAAAAAAAGI/wNlX8Szi8qg/s1600/dp2plan-crop0004.png"><img style="cursor:pointer; cursor:hand;width: 300px; height: 400px;" src="http://2.bp.blogspot.com/-V7ZN1L9BkfU/ToNW3GOwDGI/AAAAAAAAAGI/wNlX8Szi8qg/s400/dp2plan-crop0004.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657461061472750690" /></a><br />
and we get the footprint in fuchsia. Next, we will erect a rafter from point H up to the peak, with the same width and depth as the principal rafters:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/-P8X6BnkZSZQ/ToNZGdeWx0I/AAAAAAAAAGo/M4-GoTUuFLc/s1600/0005.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://1.bp.blogspot.com/-P8X6BnkZSZQ/ToNZGdeWx0I/AAAAAAAAAGo/M4-GoTUuFLc/s400/0005.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657463524433512258" /></a><br />
This rafter has plumb faces too and won't be much harder. The only problem is that we don't have an elevation view of this roof surface. So, we construct one using the height h of the roof peak:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/-JpeU3YF9-N4/ToNeMXmsqDI/AAAAAAAAAHA/uc9E6FWl8_w/s1600/dp2plan-crop0006.png"><img style="cursor:pointer; cursor:hand;width: 300px; height: 400px;" src="http://3.bp.blogspot.com/-JpeU3YF9-N4/ToNeMXmsqDI/AAAAAAAAAHA/uc9E6FWl8_w/s400/dp2plan-crop0006.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657469123495241778" /></a><br />
We constructed the elevation right over the plan. This is common in the French carpentry drawing tradition. It saves space, which is important if the plan is drawn at full scale (called an "épure".) It also allows us to easily move between the plan and the elevation. Again, the inner surface intersection line gives us the inner edge of the footprint.<br />
<p>When we set out to find the "dévers de pas" line that defines the side of a footprint, we are really looking to define the plane for that side of the timber. In order to draw the line on the plan, we need to connect two points that lie on both the timber's side and the ground plane. We usually have one point in hand already: The point where the side meets the roof gutter line. So, the problem often reduces to finding one more line in the side plane and its intersection with the ground.<br />
<p>Consider the principal rafter rising from point E. The rafter's sides are square, so if we drop a perpendicular line from a point on the top edge, it will lie in the side face:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/-oYep90fLW4A/ToNZGmXQN_I/AAAAAAAAAGw/XIWjH2aC-9c/s1600/0006.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://4.bp.blogspot.com/-oYep90fLW4A/ToNZGmXQN_I/AAAAAAAAAGw/XIWjH2aC-9c/s400/0006.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657463526819641330" /></a><br />
Since the upper surface of the rafter lies in the surface of the roof, that line is a surface normal, and is parallel to all other surface normals of that roof:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/-ldiePGjaxnQ/ToNZGpEQ1mI/AAAAAAAAAG4/eSsL68csO7U/s1600/0007.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://4.bp.blogspot.com/-ldiePGjaxnQ/ToNZGpEQ1mI/AAAAAAAAAG4/eSsL68csO7U/s400/0007.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657463527545296482" /></a><br />
<p>This is good stuff. If we need a line for our dévers de pas construction, we just need to look for a surface normal somewhere.<br />
<p>Next, erect a rafter from G up to the peak. This is our skewed rafter from the first post:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/-t9HXrygY72U/ToNh7DoDfwI/AAAAAAAAAHo/j6IRmqzw1gg/s1600/0008.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://1.bp.blogspot.com/-t9HXrygY72U/ToNh7DoDfwI/AAAAAAAAAHo/j6IRmqzw1gg/s400/0008.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657473224120958722" /></a><br />
We are not going to construct the projection line directly on the side of the footprint yet. Instead we will construct a parallel plane -- and D.P. line -- starting from G because that works better with our principal reference point, the peak of the roof at K. That plane contains the line from G to K; we are looking for one more line to completely specify it.<br />
<p>As before, we start with an elevation view of this roof surface. We can use the roof height, but the view needs to be along the roof line. We do that by running a line perpendicular to the roof edge through K and forming a right triangle with the roof peak height, taking us to point M:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/-pgkL_RuL8Do/ToNeMtF24KI/AAAAAAAAAHI/aVG7CyCevu8/s1600/dp2plan-crop0009.png"><img style="cursor:pointer; cursor:hand;width: 300px; height: 400px;" src="http://2.bp.blogspot.com/-pgkL_RuL8Do/ToNeMtF24KI/AAAAAAAAAHI/aVG7CyCevu8/s400/dp2plan-crop0009.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657469129263079586" /></a><br />
<p>If you tilt your head over to the side, you are looking at the elevation view of rafters in this roof surface. Even if a rafter doesn't rise square from the gutter line, it will usually be kept between the upper and lower surfaces shown in this view. If we cut out the elevation view and tip it up in 3D, we can see what's going on:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/-891UslrcDS0/ToNh7TZtP7I/AAAAAAAAAHw/IFFvpPRenys/s1600/0009.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://4.bp.blogspot.com/-891UslrcDS0/ToNh7TZtP7I/AAAAAAAAAHw/IFFvpPRenys/s400/0009.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657473228355747762" /></a><br />
<p>Now, draw a perpendicular line to ML at M and mark the point where it intersects our baseline KL as "Q":<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/-XY-dS13zYhA/ToNeMzK78fI/AAAAAAAAAHQ/K0MR6UNKz68/s1600/dp2plan-crop0010.png"><img style="cursor:pointer; cursor:hand;width: 300px; height: 400px;" src="http://4.bp.blogspot.com/-XY-dS13zYhA/ToNeMzK78fI/AAAAAAAAAHQ/K0MR6UNKz68/s400/dp2plan-crop0010.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657469130894995954" /></a><br />
<p>A perpendicular line dropped from surface in an elevation view is very common and is labeled "T.C." for "trait carré" or square line. Let's also flip that into 3D:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/-86BhbmNOT1c/ToNh7V2aLeI/AAAAAAAAAH4/6TwUzK_JTZw/s1600/0010.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://1.bp.blogspot.com/-86BhbmNOT1c/ToNh7V2aLeI/AAAAAAAAAH4/6TwUzK_JTZw/s400/0010.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657473229013003746" /></a><br />
<p>Now it becomes obvious why the T.C. line is so important: it's normal to the roof surface. Therefore it lies in the plane passing through G that we are searching for:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/-lJugs4KjeM0/ToNh7zX-5uI/AAAAAAAAAIA/YpwcKvMYnt8/s1600/0011.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://3.bp.blogspot.com/-lJugs4KjeM0/ToNh7zX-5uI/AAAAAAAAAIA/YpwcKvMYnt8/s400/0011.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657473236938450658" /></a><br />
<p>So we can draw the dévers de pas line in our drawing: <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/-xs7rPQGvc08/ToNeNIbZRBI/AAAAAAAAAHY/LBvqniJh63k/s1600/dp2plan-crop0011.png"><img style="cursor:pointer; cursor:hand;width: 300px; height: 400px;" src="http://3.bp.blogspot.com/-xs7rPQGvc08/ToNeNIbZRBI/AAAAAAAAAHY/LBvqniJh63k/s400/dp2plan-crop0011.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657469136601170962" /></a><br />
<p>and fill in the footprint of the rafter. We give the rafter a width along the gutter line and use the inner surface intersection from the rafter elevation, as before:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/-cRrBnCPeOvY/ToNeNI7j0ZI/AAAAAAAAAHg/4JpeJUlDm4k/s1600/dp2plan-crop0012.png"><img style="cursor:pointer; cursor:hand;width: 300px; height: 400px;" src="http://1.bp.blogspot.com/-cRrBnCPeOvY/ToNeNI7j0ZI/AAAAAAAAAHg/4JpeJUlDm4k/s400/dp2plan-crop0012.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657469136736080274" /></a><br />
<p>In the final 3D view, we see how the plane we constructed is parallel to the sides of the rafter:<br />
<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/-2TWKrc8W0DI/ToNh77n3xaI/AAAAAAAAAII/6qFhoYftUPE/s1600/0012.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://4.bp.blogspot.com/-2TWKrc8W0DI/ToNh77n3xaI/AAAAAAAAAII/6qFhoYftUPE/s400/0012.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5657473239152575906" /></a><br />
<p>This construction that uses a surface normal in the form of the "trait carré" to derive the D.P. line is used again and again in Mazerolle's drawings.<br />
<p>P.S. Chris Hall has a go at the same model in <a href="http://thecarpentryway.blogspot.com/2010/03/following-mazerolle-theorie-des-devers.html">this post</a>. Check it out; it was one of my inspirations for learning more about this French method and Louis Mazerolle's works.Tim Moorehttp://www.blogger.com/profile/09629429704217731021noreply@blogger.com6tag:blogger.com,1999:blog-1930479055984418545.post-27592753825098892862011-09-23T19:42:00.010+02:002011-10-17T13:16:33.487+02:00devers de pasI first discovered "Traité Théorique et Pratique de Charpente" by Louis Mazerolle on Chris Hall's blog <a href="http://thecarpentryway.blogspot.com">the Carpentry Way</a> in an amazing series of posts. Mazerolle was a "compagnon de devoir" or master carpenter in the 19th century in France, and his book is a compendium of complicated and obscure drawings of roofs, buildings and staircases. In this tradition the plan of a roof would be drawn out at full scale on the ground; then, the individual timbers would be placed over the plan, scribed, and then cut. 112 plates take the reader from fairly simple dormers and roofs to sawhorse challenge problems and incredibly baroque curved structures. Chris built <a href="http://http://thecarpentryway.blogspot.com/search/label/Louis%20Mazerolle%20tr%C3%A9teau">a crazy sawhorse</a> from the book and worked through several of the drawings using SketchUp, but ultimately put that aside in frustration. I was intrigued and got a copy of the tome (it's big and expensive, even in France). I started doing some of the same problems and shared my results with Chris, with the happy result that he's started blogging about it again. We are collaborating on deciphering this work and, based on our drawings, building 3D models of the structures to verify that we've gotten it right. Chris' frustration was justified. The text is extremely terse and filled with small typos, usually references to non-existent or wrong labels on the drawings. While the drawings are beautiful, the details cannot be trusted. Straight lines aren't, right angles aren't, some things are just plain wrong. I suspect that a lot of this was introduced in the process of recopying existing drawings to make the engravings. It would not be possible to build a lot of the structures directly from these plans; on the other hand, that is not the point. An apprentice would redo the drawing using the appropriate techniques and thus learn his art. When I started in on the book I realized that, though the techniques used to produce the drawings were not well explained and often obscure, they are basically sound and make sense to someone with a background in math or computer graphics. Furthermore, they become a lot more clear when one can whip up a quick 3D model to understand the 2D construction. I thought it would be interesting reconstruct some of Mazerolle's plates, with a more complete explanation of the geometry and some nice 3D graphics. This post is my first attempt at this, but it is a sort of prequel in that it tries to present the motivation for a construction, called "devers de pas," that is explained early on and then used throughout the book. "Devers de pas" literally means "footprint area," though "level section" might be more idiomatic in this context. "Devers" is actually an archaic French word that has just about disappeared, except in certain locutions, that means "pertaining to," belonging to," or "nearby." It's very close in spelling to another word "dévers", meaning "angle" or "angled," and we will actually be talking about "dévers de pas" in a bit! To top off the confusion, Mazerolle spells "dévers" as "devers" everywhere, which must have been a regionalism. The technique makes use of the footprint of a solid sitting on a horizontal plane to reason about its 3-dimensional geometry and the cuts needed to arrive at its final shape. In carpentry drawing, the solid is a piece of wood, usually some kind of rafter. However, the technique is very general and can be applied in many different contexts. As an example, here is the plan of a roof model that we see more of later in the book. Never mind the rafters poking through the surface of the roof. The top part of the plan shows the principal rafters i.e., the rafters on the sides. This establishes the height of the roof peak and the slopes of the side roof surfaces; it also indirectly establishes the slopes of the top and bottom roof surfaces, which are different from those of the sides. <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/-qIjyHFJOYLk/TnzGfRPSIYI/AAAAAAAAAEI/7wobFxAejSE/s1600/plan0-med.png"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 283px; height: 400px;" src="http://2.bp.blogspot.com/-qIjyHFJOYLk/TnzGfRPSIYI/AAAAAAAAAEI/7wobFxAejSE/s400/plan0-med.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655613472576250242" /></a> and here's a 3D view. <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/-Zm_cBDz3d-k/TnzHxu1zzaI/AAAAAAAAAEw/6tgECU_PWwA/s1600/0000.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://4.bp.blogspot.com/-Zm_cBDz3d-k/TnzHxu1zzaI/AAAAAAAAAEw/6tgECU_PWwA/s400/0000.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655614889271741858" /></a> We are more concerned with the rafters than the roof surface. <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/-SuDZcGcRm7g/TnzHyE8fDZI/AAAAAAAAAE4/aVOC-Lc-5BI/s1600/0001.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://2.bp.blogspot.com/-SuDZcGcRm7g/TnzHyE8fDZI/AAAAAAAAAE4/aVOC-Lc-5BI/s400/0001.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655614895205322130" /></a> Let's take a look at the footprint of the left (as seen from the top) rafter. <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/-A4OrEx0JuQI/TnzHysyhXiI/AAAAAAAAAFA/du-lX6ztuzg/s1600/0002.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://2.bp.blogspot.com/-A4OrEx0JuQI/TnzHysyhXiI/AAAAAAAAAFA/du-lX6ztuzg/s400/0002.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655614905900949026" /></a> Actually, we are more interested in the edges of the footprint than the footprint itself. Each edge is the intersection of the ground plane and the corresponding face of the piece of wood. So, the edge is a line that is in both the ground plane and the surface plane. This is still true if we extend the line of the edge past the boundary of the footprint: <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/-nR8R1GkaNmo/TnzHzHGXcEI/AAAAAAAAAFI/yjM5ZbSV1NE/s1600/0003.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://2.bp.blogspot.com/-nR8R1GkaNmo/TnzHzHGXcEI/AAAAAAAAAFI/yjM5ZbSV1NE/s400/0003.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655614912963506242" /></a> The extension line is called the "dévers de pas" or "angle projection" and will be annotated as "DP" on our drawings. These DP lines represent the intersection of a plane with the ground plane and, as we shall see later, can be used to draw the intersections of planes and the resulting solids. The plane of the side of a plumb rafter isn't too interesting, so let's look at the skewed rafter in the top roof surface. <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/-lPMAX_cLNZs/TnzGgUMLEQI/AAAAAAAAAEY/vgIL6_1ksaY/s1600/plan4-med.png"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 283px; height: 400px;" src="http://3.bp.blogspot.com/-lPMAX_cLNZs/TnzGgUMLEQI/AAAAAAAAAEY/vgIL6_1ksaY/s400/plan4-med.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655613490548379906" /></a> The DP line isn't coincident with the plan of the edges anymore. Here's the situation in 3D: <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://4.bp.blogspot.com/-dvVfqqo7tN8/TnzHzntr2FI/AAAAAAAAAFQ/TjICsMc2NT8/s1600/0004.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://4.bp.blogspot.com/-dvVfqqo7tN8/TnzHzntr2FI/AAAAAAAAAFQ/TjICsMc2NT8/s400/0004.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655614921718356050" /></a> In this case, the drawing of the footprint was derived from the DP line, and then the edge lines could be drawn on the plan. In the next post on the subject we will see how to construct DP lines in different situations, including this one. Sometimes the footprint is determined by other factors and determines the DP line, other times the DP line comes first. As a final teaser, we'll see how the devers de pas method can be used to make the cuts for the left face (as seen from the top) of this skewed rafter. <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/-P4CpMtf6dxw/TnzGglW-GLI/AAAAAAAAAEg/3AwKJXVOgUc/s1600/plan5-med.png"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 283px; height: 400px;" src="http://3.bp.blogspot.com/-P4CpMtf6dxw/TnzGglW-GLI/AAAAAAAAAEg/3AwKJXVOgUc/s400/plan5-med.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655613495157070002" /></a> GC is the top edge of the left face. On the plan we drop a right angle from the dévers de pas line through C, giving us the point T. We can get the height of point C in three dimensions from this drawing, but I won't do the construction now. Suffice it to say that I found the height, then drew that from C at right angles to TC, giving us U. Connect T and U to form a pointy right triangle. Also note that I've marked the intersection of the DP line with the surface of the left rafter as Z. Here's the situation if we fold up the triangle in 3d: <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/-GeeuEgH8jRM/TnzKbLyHNDI/AAAAAAAAAFY/BlCShtrG7us/s1600/0005.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://2.bp.blogspot.com/-GeeuEgH8jRM/TnzKbLyHNDI/AAAAAAAAAFY/BlCShtrG7us/s400/0005.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655617800438756402" /></a> We see that TU [correction: I had written "ZU" here] is the slant distance from T to the top of the left edge. Since that point is on the left surface, and the DP line is also in the left surface plane, The resulting right triangle is as well. We can draft that in 2D by swinging the line TU' around to be coincident with TC. <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/-jZ5DUc-UtnE/TnzGhMOu7fI/AAAAAAAAAEo/gU8Y-lhkzro/s1600/plan6-med.png"><img style="display:block; margin:0px auto 10px; text-align:center;cursor:pointer; cursor:hand;width: 283px; height: 400px;" src="http://1.bp.blogspot.com/-jZ5DUc-UtnE/TnzGhMOu7fI/AAAAAAAAAEo/gU8Y-lhkzro/s400/plan6-med.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655613505591504370" /></a> If we flip that up into 3D: <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://3.bp.blogspot.com/-RxV5L_AN_C0/TnzKbjmGoKI/AAAAAAAAAFg/gk_AamHZlCM/s1600/0006.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://3.bp.blogspot.com/-RxV5L_AN_C0/TnzKbjmGoKI/AAAAAAAAAFg/gk_AamHZlCM/s400/0006.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655617806830837922" /></a> we see that GU' is the same length as the top edge of the rafter's surface. Furthermore, the angle U'GT is the angle with the ground on that surface, and GU'Z is the cut angle at the top of the rafter. This is even more clear if we take an orthographic view of the rafter and this plane: <a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://2.bp.blogspot.com/-kDvPFamavZo/TnzKcPZrU6I/AAAAAAAAAFo/51ZzH-AoF0s/s1600/dportho.png"><img style="cursor:pointer; cursor:hand;width: 400px; height: 225px;" src="http://2.bp.blogspot.com/-kDvPFamavZo/TnzKcPZrU6I/AAAAAAAAAFo/51ZzH-AoF0s/s400/dportho.png" border="0" alt=""id="BLOGGER_PHOTO_ID_5655617818589877154" /></a> We've had a very brief look at the power of the "devers de pas" method. I hope this whets your appetite for more. Chris presents a clever use of devers de pas in his series of blog posts called <a href="http://thecarpentryway.blogspot.com/2010/11/x-marks-spot.html">X Marks the Spot</a>, which presents a challenge problem of determining the intersection of two rafters at arbitrary angles.Tim Moorehttp://www.blogger.com/profile/09629429704217731021noreply@blogger.com6tag:blogger.com,1999:blog-1930479055984418545.post-75315445052337793852011-09-23T19:16:00.002+02:002011-09-23T19:28:38.541+02:00WelcomeStereotomy is the art of cutting a three-dimensional object using a 2D plan. I'm a computer graphics guy, and I think these old methods are very cool. I'm interested in the drawings made by the <a href="http://en.wikipedia.org/wiki/Compagnons_du_Tour_de_France">Compagnons du Devoir</a> in the 19th century and will be exploring their techniques from my own point of view.
<p>
Today the term is used in architecture and construction history to refer to stone cutting and structural calculations that were made with graphical methods. I think that's cool too, and plan to take a look at the work of Monduit and Philibert de l'Orme. Basically, old technical drawings are cool.
<p>
"Stereotomy" is also an album by the Alan Parsons Project, but I won't be talking so much about that.Tim Moorehttp://www.blogger.com/profile/09629429704217731021noreply@blogger.com2