In the last post, I introduced Plate 1 in the text La Menuiserie, specifically volume 3 of aa multi-volume encyclopedia on the topic. Plate 1 detailed the first significant exercise in a chapter entitled Eléments de géométrie dans l’espace. The study in this section concerns the intersection of various solids, like cones, parallelepipeds, cylinders, and spheres. I’ve worked my way through all of the remaining exercises in this section, and was pleased to find the layout techniques worked perfectly. In fact, the technique for determining the intersections between solids is fairly uniform between different cases, with one exception, to be mentioned below.

Plate 2 covers the intersection of a sloped square section stick piercing a cylindrical post:

Plate 3 deals with the intersection of a sloped smaller cylinder meting a cylindrical post:

Plate 4 gets us into cones, the first problem dealing with a cone pierced by a horizontal cylinder:

Plate 5 is a cone pierced by a smaller vertical cylinder, offset from the centerline of the cone:

Plate 6 was the most complex of this set, involving the intersection of an angled cylinder partially occluded into a cone:

Plate 7 was a different sort of beast, the intersection of a vertical and a horizontal cone:

Here, one can not pull points in the same way as with other intersections, but must construct auxiliary planes which intersect both cones.

Detail:

If you’re hankering to see some sort of woody manifestation of the above type of problem, maybe a bit more difficult, volume dialed up to 11, there are no shortage of maquettes. This one could be called, “let’s have fun with scalene cones”:

The preamble to a carpentry challenge like this – the minimum ‘get a foot in the door’ move – would be mapping the surfaces of intersecting cones.

It appears that the upper angled cone can be removed easily:

Scalene fun, Part II – it is the same model, taken off the support stand:

Credit to: RWLV

Finally, there is plate 8 which deals with a sphere occluding a cone:

This was a simpler problem than the cone meeting cone, as it turned out. Almost easy.

The next section in the text deals with regular plan hips, and area with which i have some familiarity, so it will be interesting to learn some new approaches.

All for this time. If you’re feeling inspired – and I hope you are- go lay down some lines and cut some wood. Post IV is next.

## 7 thoughts on “La Menuiserie Study (III)”

1. I'm reaching for a bottle of aspirin. Funny, but I can actually see the relationship between the drawings and the objects and my spatial abilities suck.

2. Ralph,

see? It's not so bad after all. Have no fear though, the slope you teeter on is not a slippery one – rather the opposite, and uphill in nature.

~C

3. Chris,

Do you have a background in engineering? You're more of an accomplished geometry geek than I am, and I'm 3 years into a 4-year engineering program 😉

In all seriousness though, I'm appreciative that you're doing g these exercises for all to see; my woodworking mentor and I were talking about the tendency for woodworkers to remain in the rectilinear their entire careers…probably because curved shapes and their intersections are difficult! Watching you break these down inspires me to explore “out of the box” designs (pun intended !).

–Josh

4. Hi Josh,

I'm glad you are enjoying the series. I wasn't sure it would be of interest to many folks, so it is rewarding to hear that you found it worthwhile. I agree, the vast majority of woodworkers never venture outside orthogonal construction, and it is to the detriment of the development of the art. Orthogonal will always be the vast majority of the work, but there are delights to be found once you venture beyond those boundaries.

~C

5. Cylinder/cylinder intersections used to be one a standard drawing exercise for mechanical engineering students.

This is one area where I'm glad we have 3D CAD now.

6. Roland,

yes, I hear you. If these type of drawings were something I had to do a lot, the manual method would become tedious after a while. The method is interesting to learn, and play around with, and after a while the drawing development is really pretty much the same operation for each type so it becomes less interesting.

In sketchup there is no function to enable a surface to be rolled out flat after intersection, so 2D development remains the best option for me.

But, again, if I had to do this sort of thing with frequency, I'd be looking into a more suitable CAD program.

~C

7. While lots of CAD programs have the ability to create 2D views from 3D parts, the closest I've seen to “unfolding” a part is in sheet metal software.

Come to think of it, some 3D modelers do a kind of unfolding for texture mapping. In that case though local stretching and compression is explicitly allowed.

But as you undoubtedly realize, only a fraction of shapes can actually be unfolded. And a lot of manufacturing these days goes straight from 3D to a CNC mill, so I wonder if there is enough demand for a feature like this.

Next to real 3D programs like Siemens NX and Creo/Parametric I still use AutoCAD Light for accurate 2D sketching. Something like that would certainly help to make the manual method faster and more accurate. Using such 2D drafting software in combination with a plotter is very handy.

Just recently I needed a fixture to position some metal inserts in a composite structure for bonding. I made a 2D sketch, plotted it 1:1 and fastened the plot to a piece of plywood with double side adhesive tape. I used a band saw for the contours and a center-punch and drill press to drill accurate holes for the screws to hold the inserts at the indicated locations. It worked perfectly.