Bombe Vitrine

Compound Curve Cope and Stick Router Techniques

'Bombe' is a French term, it means 'rounded' and has been used to refer to a variety of furniture designs. 'Vitrine' is French too, it comes from the Latin 'vitrum' meaning glass. A vitrine is a fine glass display case.

Have you ever had an idea that you couldn�t shake off, demanding that you give it a try? That�s what I went through for many years before undertaking this rather time-consuming project. From the start of my woodworking career I was more interested with curved work than rectilinear, and this project carries that interest to one of its further of possible manifestations. This project began with a desire to produce a rounded, organic form not limited by the rectilinear requirements of standard woodworking tooling. Though the project carried me far afield with geometric abstraction, my original intention was not to create complex techniques purely for their own merit, intriquing as they may be. The intention was to make a fairly simple form, but it so happened that the techniques required became rather involved.

It�s fairly easy to put a single curve into a woodworking design, such as a curved top rail on a flat cabinet door. Going one step further, you can fairly easily bow the door, that is, make it curved back and forth but still straight up and down, requiring single bent glass. Note, though, that if you both bow the door and put a curve into the top rail that the top rail now has two curves in it, and qualifies as a compound or complex curve. You�ll find fascinating treatments of techniques for this kind of work in "Circular Work in Carpentry and Joinery", by George Collings. Click on the book name to see more about it.

A bowed door, with a curve top rail or not, has a cylindrical shape, which is to say that you can think of it as a section of a larger cylinder. The next step geometrically is to move from a cylinder to a sphere, as does my bombe vitrine. The glazed frames on this cabinet are sphere sections, which proceed from specific radii. Other more complicated forms are possible (some of which have been done) which involve either free sculpting or more complicated geometrical designs such as ellipses or differing curves on different axes (a sphere, by definition, has the same curve on all axes).

One general rule applies. The more complex your design the more time consuming and difficult it will be to make. The bombe vitrine shown here took a total of about 1350 hours to complete. If you want to take on a project like this, settle in for the long haul- you won�t be done by Christmas.

This treatment of the processes I went through to build this cabinet does not contain specific instructions for building the cabinet shown. I�m not trying to protect the design, but I doubt that anyone will try to make a duplicate of this one and you probably don�t want that level of detail. This treatment just shows you the major techniques used in spherical cope and stick router work, and points you toward the complexities involved with designing such a piece of furniture. If you really intend to build something like this, contact me and I�ll try to persuade you not to. For your sake, believe me. If you persist, I�ll help you with design specifics to a certain extent if you are easy to work with. I�d like to see someone try this, but not if they end up frustrated.

 

WHEN WORLDS COLLIDE- THE PLANE OF INTERSECTING SPHERES

When two planets smack together, they explode. I know, cause I saw it on TV once. But, when two imaginary spheres come together, they intersect in three dimensions like two large soap bubbles joining together in some sort of ethereal geometric bliss.

To understand what it takes to assemble a construction designed as sections of spheres, you must be able to visualize what happens when spheres intersect (see drawing 1). You must know where the radius center of each sphere is in relation to the other sphere center. To figure how I would join the sides of this cabinet, I made a full scale section drawing of a horizontal plane through the widest part of the upper cabinet as in drawing 2. (The sphere centers lie on this plane.) This drawing forced me to decide which radii I would use for the piece, and where the sphere centers were located on the X and Y axes. Later alignment of the outer frame parts as they were made in the router jigs depended entirely upon this drawing and others similar to it.

Think about those soap bubbles again. Floating apart in the air, each is a separate sphere. But when they contact, a flat area of soap membrane resides within the circle that is created where the two spheres contact. If the bubbles are true spheres, the circle of contact will be a true circle, and this circle will describe a flat plane. This is the plane of intersecting spheres and is the only flat surface you have to refer to in design and construction of such a piece.

Note that the horizontal plane I used to draw the X and Y axes, the sphere centers and the cabinet parts (drawing 2) is not the same plane as the plane(s) of intersecting spheres. At each of the four corners of the cabinet, and at each of the four sides of the top where it intersects the sides below, a different plane of intersecting spheres is described. Each of these planes must be plotted in relation to the centers of the spheres it involves. You must know where the planes of intersecting spheres are in relation to the sphere centers and cabinet central axes in order to be able to align components in the router jigs for shaping. One critical piece of information I used is the fact that when you are dealing with intersecting spheres of equal radius, the plane of intersecting spheres must pass through the midpoint of a line drawn between the sphere centers.

Confused? Well so was I, and for months. I racked my neurons for weeks over some of these three dimensional convolutions, and spent a few sleepless nights where I finally realized a solution at 3AM, thus allowing me to sleep. Finally, however, I concluded that it is really very basic three-dimensional geometry- just a bunch of spheres and planes bumping into each other at specific angles and with specific radii. The problem is that the human brain prefers two dimensional, linear problems rather than three-dimensional curve balls. But, once you get used to visualizing the parts involved you�ll find that the basic geometric relationships are simple. Making drawings helps a great deal to conceptualize what�s happening, as well. 

 Though it's possible to make many of these geometric situations very complicated with algebra and calculus, it's also unnecessary. My approach was pragmatic and physical- I did what I needed in order to make the router move along the shape required. I used geometric abstraction only as far as I needed to in order to learn how to arrange my jigging such that it would produce parts that fit. I was not so much concerned with making the cabinet parts fit a rational abstraction perfectly as I was determined to make the cabinet parts fit each other practically.

BUILDING THE JIGS

When you use a router table with flat stock, you use a flat router table. When you do router work on spherical pieces, you need spherical surfaces to place your work upon. With a flat router table and flat work, when you want to flip the part and put the opposite face on the table you use the same table. With spherical parts you must make two jigs- one that is domed and one that is dished. You work on the outside of spherical parts on the dome jig, and work on the inside on the dish jig. See photos ahead to look at these jigs.

These jigs are somewhat time consuming to make but you will spend far more time using them than making them. The ultimate accuracy of your work will be a function of the accuracy of your jigs, so it is wise to build them well. But, know where you must be accurate and where it doesn�t matter. The accuracy of the dome and dish jigs depends on how well the spherical surface is cut into them, that is, how close it is to a true sphere. As well, the accuracy of these jigs depends upon how rigid the structures are that suspend the router above the work. Great care in locating the pivot points for the swinging template arms and making them and their supports rigid is necessary. It is not necessary to make the dome and dish glue blanks all that accurate up to the point of glue up. They just need to be close enough to the required shape that the router can cut the required sphere section into them. They are carving blanks.

The flat surface beneath the dish jig needs to be flat and rigid, because special jigs that hold parts during the first shaping cut must be located on this surface. The surface to which the dome jig is attached doesn�t need to be flat because the only thing that gets attached to it is the dome jig itself. However if it is terribly out of flat you will have difficulty aligning the pivots for the template arms

Use a stable wood for the dome and dish jigs, so that once you have cut out the spherical shapes into them they will be less likely to distort with moisture variations in the air. I used recycled redwood fencing for mine. This old growth redwood, pulled out of the San Jose, Calif. landfill, is very stable stuff but it had a lot of knots in it. Since I was sinking numerous screws into the jigs to attach parts the knots became a problem occasionally. One screw sheared off in a knot, then I forgot to pull it before resurfacing the jig with a large router bit, which was damaged when it hit the screw. This was the only time in the entire project that I hit metal with any router bit- a good record given how close the expensive cope and stick bits come to screws during use.

Back to conceptualizing about spheres so we can figure out how to make the pieces that go into the dome and dish blanks. Imagine a four foot diameter sphere with a flat, horizontal plane passing through the middle of it (drawing 3). Now, imagine parallel planes spaced at 3 inches from the first, both above and below it.

Continue adding such planes until there are enough to fill the inside of the sphere. Note that the first plane contacts the sphere at a tangent that is 90o to the plane. But the planes adjacent to the first plane contact at a tangent with an

angle less than 90o. The further each plane is from the center plane, the less the angle is of its tangent to the circle. From this we see that each successive layer of the glue up lamination for each of the dome and dish jigs must be cut at a different angle than the last.

As well, each successive layer must be cut at a different radius, because the circle described by that layer has a smaller circle radius to the Y axis, even though that circle has always the same sphere radius as the other circles to the sphere center.

I began my jigs by gluing together three pieces of 1x fencing as in photo 1. First I planed the fencing so it would glue well and to make the pieces uniform in thickness. After laminating the groups of three together, I edge glued them together to make large plates as in photo 2. I made a full scale section drawing through each jig to show me the angles and radii required for each as in drawing 3, then began scribing the radii onto the large blanks with shop-made trammel points as in photo 3. Next came band sawing each of the laminations out as in photo 4. For efficiency I glued the off-cuts onto other blanks as in photo 5.

 

Photo 6 shows the first glue up for the dish jig. It gets glued up in an upside-down position so that the critical inside surface can lie on two special plywood reference jigs during the glue up. Those jigs hold each of the laminations in correct relation to each other so the end result approximates the sphere required. The radius of those jigs is a function of how far apart they are from each other, just as the radius of each lamination is a function of how far they are from the central plane, as explained above.

Photo 7 shows the first glue up of the dome jig, again upside down so that the surface of the jig that we want to use is the one that contacts the reference jigs. After that dried I glued on two more laminations on each side, after that dried two more went onto each side as in photo 8. I made two domes, one for the router jig and one to use as an assembly table to glue the frames together on.

MAKING THE TEMPLATE ARMS

You�ll see from the photos that the routers sit in little carriages that ride on curved templates with small bearings. The radii of these templates is greater or lesser than the radius of cut required, by however much was required for the router carriage. For rigidity I made the templates six inches wide and used �" thick birch plywood. I swung the arcs on the templates using a router arcing setup as in photo 9. To do so you must very carefully measure the distance from the pivot point of the arcing jig (just a nail or screw) to the bit as in photo 10. Measure to the outside of the bit for an inside curve, and to the inside of the bit for an outside curve.

It�s only necessary to use the arcing jig to make one of each size of template. Then use each of these to trace the others you need and cut them out on the band saw as in photo 11. Flush trim the duplicates to size on a router table with a bearing guided router bit as in photo 12. Smooth the surfaces that the bearings will ride on with sandpaper and a flexible sanding block as in photo 13.

Your goal is to cause the router bit to swing in a specific sphere radius on the jigs. So, you must suspend the bearing templates above the dish or bowl with extensions that cause them to pivot from the correct axis. The line created by joining the two pivot points on each jig must pass through the sphere center that you are establishing. Photos 14 and 15 show how I set up each jig. Note that they are mounted atop shallow plywood boxes. These are storage crates I�d made for something else; they were convenient for this purpose due to their rigidity.

 

 

A full scale drawing was required for lining up the relation between the curved templates and straight arms attached to them. The length of the straight arms is a function of how far they are from each other and the center of the circle. The full scale drawing shows where to align all parts to ensure that the curved template edges are the same distance to the sphere center along their entire length. Such a drawing is similar to drawing 3, where you are locating planes at given distances from the sphere center and discovering where those planes intersect the sphere.

Once the templates and arms are all together, it�s necessary to surface the dome and dish to the correct radius. The dome gets cut to the inside radius of parts, the dish to the outside radius, so that when parts sit on the jigs they sit "flat" and don�t rock.

 

	Photo 13- The surface on which the bearings ride needs to be very smooth.
	Photo 14- Dish jig. To the right of the router is a holder jig with a frame component mounted in it. This piece is getting its first finished curved surface cut into it. On the left of the router is the dish glue up, on which parts will be screwed for later operations, specifically rabbetting and mortising. Parts can't go on this dish glue up at first, though, because they don't yet have the curve cut into their tops to match the curve of the dish.

MAKING SPHERICAL SURFACES ON PARTS

The first pieces of walnut that I actually cut were the outer frame parts for the five frames that make up the glazed portion of the cabinet. Vertical frame components are called stiles, and horizontal are called rails. This doesn�t apply for the top since all of its parts are horizontal. Rail ends, however, butt onto the edges of stiles whereas stile ends run full frame dimension.

The first curved cut made on each piece was to dish out the inside as shown in photos 14 and 18. But it is essential that this cut be made in correct relation to the plane of intersecting spheres. We�re back in abstract geometry world for this. Note that there are a total of 8 corners on the cabinet where one spherical frame intersects another. There are 4 on the cabinet sides, and 4 where the top meets each of the sides. Each of these corners represents two spheres intersecting, and therefore has a plane of intersecting spheres. This flat plane is where the parts on the cabinet join. Note that there are, therefore, 8 separate planes of intersecting spheres described by the design of the cabinet.

 

Measuring the angle made between the plane of intersecting spheres and a tangent to the sphere that proceeds from a point where the sphere contacts the plane is one way to locate the relation between the plane and each sphere. But it is difficult to implement the cuts requir

ed using this method. The better way to go is to determine where the plane of intersecting spheres is in relation to the main axes of the cabinet and the sphere centers, and delineate where the part resides on the plane in relation to these. Each plane of intersecting spheres has a unique angle relation to the axes of the cabinet and can therefore be measured in relation to them in terms of the angle itself and where it intersects the planes of the axes, or planes parallel to them

Look at photos 14 and 18. The walnut part is screwed to a flat piece of plywood on a holder jig. The plane to which the part is screwed is the plane of intersecting spheres for that particular part. That piece of plywood is set at a given angle to the flat base of the jig. That base is a plane at a given distance from the sphere center and corresponding axes of the jig. That sphere center and the cabinet axes are drawn on a full scale drawing (drawing ), showing their relation to the plane of intersecting spheres. The full scale drawing, with a drawing of the holder jig and jig base plane on it, shows you just where to locate the holder jig on the base plane such that the dish cut is made correctly.

Confused? Remember, it took me a long time to learn to conceptualize this stuff. The best way to learn it is to do it. Make some full scale drawings. You�ll have fun, I guarantee it.

To make parts for the frames, I first face jointed and then planed stock so that both faces were flat. One of these flat surfaces will remain on each part and become the surface that joins the adjacent frame on the plane of intersecting spheres. Then I made templates showing the curves of the parts along the planes of intersecting spheres. Computing the radii of these templates is like computing the radii of successive layers of the jig blanks. Using these templates I traced lines onto the flat stock as in photo 16, then cut out the parts on the band saw as in photo 17. I cut the parts out over size so that the router had something to cut, and cut on the band saw at the required angle as shown on my drawings.

Next each part goes onto the dish jig on its own holder jig as in photos 14 and 18. I didn�t need a separate holder jig for each part, since some are duplicates of each other. With each part in the holder jig I then had to be sure that the router bit in the dish jig was lowered to a height that would produce the correct radius on the dish cut. After all this preparation I finally began to cut dished surfaces in walnut parts.

After the parts were dished, they could go onto the dome jig as in photo 15. This cut brings the parts to thickness. Note that in this step and many succeeding steps you don�t have to think about the plane of intersecting spheres and all that stuff. The basic relation between the plane of intersecting spheres and the curved surfaces was established at the dish jig in the first step. All else follows directly from that dish cut until you get to joining the four corners of the frames.

CUTTING PART EDGES

It would help a great deal if you understand how cope and stick joinery is done on a flat plane before you tackle it here on a sphere. So I recommend that you read another page on this site first, which covers this subject. There�s no cost for it and you can link to it here.

After the parts were brought to thickness, the next step was to flush trim the inside edges of the parts on the dome jig as in photos 19 and 20. There must be a mathematical means of establishing the radius of curvature that you need to draw onto the flat plywood template stock in order to make a template that, when pulled onto the curve of the parts, will make a flush trim cut that is parallel to the other side of the part. But I don�t know what this mathematical means is. I pulled the plywood across the part and traced the curve from the edge of the part, then cut on a band saw and sanded smooth. This was adequate, but with a true radius you could cut out the template with a router arcing jig and get a dead true result.

The plywood I used is called �wiggle wood�. It is a special plywood that bends easily, used to make single bent cabinet shapes.

The flush trimming cut on the edge of the parts makes the inside edge be 90o to the faces (not exactly- more on this later). I did it in two passes, since flush trimming a whole inch thickness at once was a bit too much to do without chattering. Photo 19 shows the first cut with the router bit riding on the template, 20 shows the second cut with the bit lowered and the bearing riding on walnut.

The next cut was the sticking pass, photo 21. The sticking is the molded edge on the inside of the parts which forms the rabbet that the glass sits in. See drawing # . The router bits I used to cut the sticking and corresponding cope were custom made to my specs, and cost about $500. I may have been able to use stock cope and stick bits, but I didn�t like the way they looked and anyway they are probably too large to fit on the dome jig without smacking into the jig itself during cuts. My bits were made by Lemmon and Snoap, a manufacturer of custom router bits for industry. If you have bits made, you must make a scale drawing specifying all radii and dimensions.

After cutting the sticking on all parts, I took them back to the dish jig for the rabbetting cut, photo 22. Before doing so, however, I set up the cope cut on the dome jig to make a few cope cuts to use in setting up the rabbetting cut. The height of the rabbetting cut determines the height of the sticking, which determines the fit of the sticking in the cope cut. It was necessary to carefully adjust the exact height of the rabbetting cut to ensure a good fit of the sticking on the cope.

JOINING THE FRAME CORNERS

Can�t let you go too long without getting into the abstract stuff, but that�s why you�re here, right?

The next step is to make the end cuts on the rails, which will determine the size of each frame. To determine the locations of these end cuts, you must make full scale drawings of the separate frames on the assembly dome (the second dome). To make these drawings, you must know the size of each frame. The best way to refer to the size of these frames is by the outside corners of them, that is, you must know where the outside corners are in relation to the cabinet axes. How do you do this, when there are no rectilinear references to measure from?

This problem was, for me, the most difficult conceptual aspect of building the cabinet and kept me awake for several nights. I�ll do my best to explain it here, and I hope not to confuse you. It took numerous drawings and a lot of head scratching before I finally cracked this nut. To understand this I suggest you make drawings too, read the text, refer to both, then write me and tell me how I should have explained it to make it more comprehensible. If your explanation of how my explanation was lacking is, itself , less lacking than mine, I�ll consider posting some or all of it here.

THREE VARYING FACTORS MEAN CALCULUS

I greatly simplified this situation by arbitrarily deciding how high above the cabinet base the top corners would be. There are four top corners, where the sides and top meet. All four of these points lie on a horizontal plane, which is parallel to the table top base that the curved sides sit upon. By arbitrarily deciding that the distance from this plane to the base was X, I avoided having to calculate where this plane was, given the height of the cabinet at top center.

On a rectilinear cabinet the height of the cabinet and the height of the corners are the same number. On a spherical cabinet they aren�t. The top of my cabinet is higher than the corners due to the spherical shape of the top. Two other factors contributing to the location of the corners are the locations of the two sides relative to the top, and each other. So, the location of each corner is determined by how three sphere sections intersect. If you know the radii of the spheres and the relation of all three sphere centers on the axes, you must be able to use calculus to determine the corner location. However, though I think I passed calculus in high school, I�ve forgotten it all long since.

By eliminating one of three factors I reduced the mathematical complexity of the problem, making it more accessible to my feeble linear thinking. I determined the radii and sphere center locations for the sides, but only the radius for the top and not its sphere center location. (All sphere radii for all sides and top are the same on this cabinet.) Then I decided how high I wanted the corners to be. In effect, I was letting the top center end up wherever it would given the fixed height of the four corners.

The next question is- where are the four corners on the horizontal plane that they all lie upon. This I determined by making a drawing of that horizontal plane, and scribing onto it where the arcs of the sides intersect the plane. The intersections of these arcs with each other locate the top corner points. The radius of curvature for these arcs that intersect the plane is not the same as the radius of curvature of the spheres themselves. This is because the plane of top corner intersection does not pass through any of the sphere centers for the sides. All four of those sphere centers reside on another plane, which is 16 inches below the top plane and constitutes the X axis-plane of the cabinet.

We�re dealing with the same situation as with the pieces constituting the dome and dish jigs. Each piece had to be cut on a different radius, depending on how far the plane of the piece was from the sphere center.

Imagine one cabinet side, then imagine the entire larger sphere that this side came out of. Imagine a horizontal plane passing through the sphere center, then imagine a parallel horizontal plane 16 inches above the first. This is the plane of the top corners, and it intersects the sphere along a circle. The radius of this circle is less than the sphere radius because the plane is a certain distance (16") from the sphere center.

We can determine the exact radius of this circle with a Y axis drawing of one of the cabinet sides. This drawing will show us how much less is the distance from the sphere to the Y axis at the corner plane than at the X plane.

Note that the same procedure for determining the locations of the top corners works to find the locations of the bottom corners which lie on a real plane- the base table.

MEANWHILE, BACK AT THE RANCH

Now where were we... oh yes, we had just made all the frame parts, put sticking and rabbets on them, and we were about to do the joinery at the rail and stile corners when we were distracted by some nasty geometry.

When you have made the drawings of the top point plane and base plane, and located on them the frame corners as we saw above, you now have all of the information you need to draw the location of parts onto the assembly dome. Making these drawings on the assembly dome is really the only way to align the parts in each frame. I made these drawings using large shop-made trammel points to function as a compass. The drawings of the top and bottom planes showed me the linear distance between corners, so with the trammel points I could draw arcs that were always a given linear distance from the trammel center, even though I was drawing on a spherical surface.

Photo 23 shows a part of the alignment procedure. After I drew on the four corners for the cabinet top, I screwed down the two stiles, as shown, on the lines. During the procedure I needed to transfer marks from the dome to the top of the stiles so I could place the rails on top of the stiles as in the next photo. But to transfer the marks I needed to know that I was extending a true radial line. The little jig in photo 23 does that- it simply holds the end of the short piece of yellow wood at 90o to a tangent at the dome surface, so that, by definition, the end is along a radial.

My dog likes to read hub-caps and therefore is frequently along a radial too, eschewing bias ply with an indignant bias.

What all of this alignment is leading up to is shown in photo 24, where the rails are placed on top of the stiles and used to scribe onto them the line at which the stiles will eventually be cut off. When this was done I removed the stiles and screwed down the rails on their lines, then placed the stiles on them (like photo 24) and scribed onto the rails where they would be cut off. Thus, final alignment for determining the frame sizes was done by physical scribing on the pieces in this manner, rather than abstractly measuring their length.

SHORT PHILOSOPHICAL DIGRESSION

Figuring part lengths abstractly on a project like this is possible but very complex because you would need a means of measuring along a curved part to a point where another curved surface intersects. This can be done, but it�s easier to keep all the abstractions in your head and use them to guide concrete scribings. In the old days woodworkers used very few abstract measurements, relying on story rods and other physical means of measure with no numerical increments involved. We, with our measuring tapes and digital calipers may get the idea that everything needs to be referred to a third party measuring system like inches or millimeters. But since your object is to make the pieces fit rather than to make them fit an abstract schema, you simplify the procedure by abandoning a third party measuring system where it is not necessary to make things fit. I�m not saying throw away your tape and calipers, they are time savers in many situations. Just be willing to put them down when they are not.

END OF SHORT PHILOSOPHICAL DIGRESSION

You may be relieved to know that we are past the majority of the abstract conceptualizing in the project. Once you have made the parts for the frames and established the length of the parts by scribing on them, your project is entirely committed to the specific design you have developed. You can�t change the part lengths without altering the way they all fit together. This is often true in rectilinear woodworking, but often not true, too. For instance, you can decrease the height of a rectilinear cabinet without affecting the width of the sides, front and back. Can�t happen with a bombe vitrine. Decreasing the height of this cabinet (lowering the top) either forces you to increase the width of the sides at the top, or decrease the overall width of the sides.

Okay, sorry, I said we were past the abstract stuff. The point of the above is that once you commit to a design you must stick with it throughout. The good thing about this is that once you have all this established as outlined above, you can leave the abstractions behind and just have fun doing compound curve cope and stick on the frame parts and tree branches. The bad thing is that if you blew it on the abstract stuff, you are in for a surprise when you go to fit all the frames together. As you can see from the photo, that didn�t happen with me.

Getting a good scribe on the frame parts is your last operation that affects the fit of the five frames together. It�s your last chance to double check your design and the relationship between each of the sides to each other and the top. Misalignments now (or that exist now from previous errors) will result in the flat portions of each frame part failing to be aligned correctly to the appropriate plane of intersecting spheres. The result of this is that the frames won�t fit- there will be tapered gaps where the flat parts should mate each other.

MAKING THE FRAME JOINTS

Having scribed the locations of the ends on the rails and stiles, the next step was to flush trim the ends of the rails, as in photo 25. The first thing you�ll notice about that photo is that the tail end of the rail that is clamped to the dome jig is off the jig surface. It is not lying flat on the jig, but is tilted up. This is intentional, for the following reason.

Refer to drawing 6, which is exaggerated for clarity. This shows the flush trimming operation on the dome jig. If the jig is built correctly, the center of the flush trim bit will point toward the sphere center of the dome jig. But, the cutter on the bit is a certain distance away from the center of the bit. Therefore, the surface cut by the bit into the part is not along a radial, which is to say that it does not point toward the center as the drawing shows. So, when you flush trim two surfaces this way and then try to mate them as we are trying to do with the rail ends against the stile rabbets and sticking, the surfaces will not mate because they are not aligned to center, as shown in the second part of the drawing.

This is an unfortunate consequence of the fact that I chose to use router bits meant for rectilinear work with spherical work. The mismatched joints perplexed me for a while, until I saw the geometry. Then I realized that I either had to have customized bits produced with flutes cut at angles that matched the geometry of my jigs, or tilt each piece as it was cut so that the angle of cut matched. I experimented with the latter until I found a way to do so reliably, which is shown in photo 25. There is a small spacer under the rail, at five inches from the end being cut. So long as this spacer was always at five inches from the end, it angled the piece just right for a correct fit. All pieces that got a cope cut into the ends had to be angled in this fashion, including all tree branches.

You might think that tilting parts this way is an unreliable and inexact means of obtaining tight joints, and the fact is that it is not an exact science. But I realized that having customized bits produced would have been problematic. Though they would have been geometrically correct in theory, the fact is that there was variation in the radial alignment of the rabbets and sticking to which the rail ends and tree branch ends were being fitted. These variations were due to the fact that the jigs were accurate but not absolutely precise. As well, when I cut the sticking on a lot of the tree branches (to which other tree branches would be fitted), these small pieces were difficult to make lie exactly flat on the jigs, causing variation in the rabbet alignment. Not much, but enough to affect the fit of copes against the sticking. By varying the location of the tilt spacer for individual joints, I could accommodate all such variations. The end result is that all cope joints on the piece are very tight.

I think this is the only way to do such work unless you build far more rigid and exact jigs that will produce parts that are dead on geometrically correct. But at the same time, why bother making such expensive and precise jigs, when it�s easy to custom fit each joint to accommodate small variations? It would take far less time to make such accommodations than to build "precise" jigs. Even if you did build jigs that were "precise", Murphy�s law might make a fool of you and you�d end up custom adjusting the angles of cope joints anyway. (Murphy's law- if it can go wrong, it will). Once again, the goal is to make the joints fit tightly, not to make the joints fit an abstract schema precisely. To my mind, the minimum technology required to do so is all that�s necessary. After completing the procedure I was satisfied with the degree of accuracy that the jigs were capable of producing as built. I was willing to make accommodation for the extent to which they were not absolutely precise, because that accomodation was not excessively difficult or time consuming. As you�ll see, each joint is individually cut anyway, so adding the tilting procedure was not burdensome.

Back to photo 25. Note the small plywood flush trim template on top of the part. This, like the flush trim template used to make the sticking edges, should have a curve in it that matches the curve of the stiles on the frames. This template is attached to two blocks, one on either side of the rail being worked. These are cope blocks. They do numerous things; they give something to screw the template to, they back up the cut on the part to reduce tearout, and they give a surface for the cope cutter bearing to ride on as it enters and leaves the part (since it doesn�t ride on the template like the flush trimmer).

Important things in this procedure are to get the flush trim template very accurately located on the scribe mark made on the rail, and to get the cope blocks tight. The part being cut has sticking cut into it on one edge, so the cope block that butts against this edge must have a cope profile cut into it, and this cope profile must fit the sticking so tightly at the point of cut that it pinches it to prevent tearout. You�ll see more on cope cuts when we get to the tree branches (look ahead at those photos).

SPHERICAL LAPPED TENONS

I considered using dowels to join the frames- for about three seconds. Yes you could make special dowel jigs that align the holes correctly etc., but I don�t want to be the one who does it. The lapped tenons shown here could be made with the jigs as they were, and a tenon beats a dowel any day. Each joint consists of three pieces lapped together. The tenon is a loose spline that bridges the joint and joins the pieces, the two pieces on top of the tenon simply cover it over to support it and complete the structural integrity of the joint. It was necessary to lap the pieces because I could not cut a spherical mortise without buying a customized router bit manufactured at the correct radius. That is, I suppose, a viable option. The laps, however, were easy to make (if time consuming) and worked quite well.

The procedure required cutting two levels of mortises, the lower for the tenon itself and the upper level for the lapped laminations that cover the tenon. Photos 26 and 27 show the flush trim template used to cut these mortises.

A little hand work was required to complete the joints as we see in photo 28 where a little wall must be scribed to the line of its adjacent lap and then removed by hand with a chisel.

The tenons themselves were made with procedures very similar to those used to make the parts for the frames. First each tenon was dished out to the internal radius, then placed on the dome jig and brought to thickness. Then at each joint the three pieces had to be custom fitted to the mortise edges and shoulders. The following photo sequence (photos 29-33) shows how the joints go together.

Photo 29 shows the completed joint ready for assembly. Note that the joinery has removed most of the cope joint, but that it does remain on the inside corner where it will be seen.

Photo 30- First the rail is butted up against the edge of the stile with the cope overlapping the sticking.

Photo 30 shows the cope of the rail brought onto the sticking of the stile.

Photo 31- Then the tenon is put in place, bridging the joint between rail and stile.

Photo 31 shows the tenon in place in the lower mortise.

Photo 32- Next the first of two top plates goes in place to cover the tenon.

Photo 32 shows the first top piece placed over the tenon and rail.

Photo 33- The second of the top plates in place, completing the joint.

Photo 33 shows the second top piece in place over the tenon and stile.

GLUE UP

Each frame was then glued up on the assembly dome as shown in photo 34. I wetted out the tenons and laps, put it all together on the drawn lines on the dome, and clamped it together with special screw blocks as shown. When out of clamps, I brought the tenons flush with the rail and stile edges on the jointer, a procedure for experienced jointer users only. The big test came when I took the five frames and taped them together to see how well the edges were aligned to their respective planes of intersecting spheres. All were close, few were exact, and I ended up using a block plane for final fitting. But they were close enough that this fitting was not too difficult. That was a big relief.

MAKING TREE BRANCHES

The nice thing about making the tree branches that fit in the frames is that there are no planes of intersecting spheres to worry about, or other complex geometrical issues. All of that is taken care of by the jigs- if they are set right and you know how to use them it�s just a matter of making parts. This is nice because it leaves you to focus on the most visible aspect of the finished cabinet- the fit of the cope and stick joints between the pieces constituting the branches.

I began with an outing one fine sunny day to photograph the branches of native oak trees here in California, specifically at the University of California at Santa Cruz, my alma mater. Using slide film I got as many shots of well-shaped branches as I could, then using a slide projector I put these photos up on the wall. Next I taped large pieces of paper to the wall and began tracing out possible forms. My first set of drawings was very bad- too many branches and not enough glass. It is, after all, a display case. The next set is what you see, more glass and carefully proportioned trees and branches. I didn�t trace whole trees from the slides, rather I patched bits and pieces here and there, as well as making my own sketches where necessary. But Mother Nature is a better artist than I and she is author of most of the branch shapes.

Next I used my final set of drawings to trace out templates for each branch. Placing the paper on top of wiggle wood plywood I punched through it with a sharp nail along the lines, then removed the paper and traced along the nail holes. After band sawing out the piece I took it to the drill press with a small drum sander and smoothed the edges a bit.

Each tree branch started as rough wood on the dish jig. Since there is no plane of intersecting spheres to worry about, the first cut is not a flat plane as with the frame components. I needed to glue up some pieces first, though, to get the curves needed for longer parts. These glue ups were all guesses in terms of how much wood to use and at what angle to glue them, but they all worked out.

Photo 37- One side of the tree branch getting fluzh trimmed to the shape of the template.

Such a glued-up piece is shown in photo 35 in the dish jig. It�s held in place just well enough to establish the dish surface. Then it moves to the dome jig as in photo 36 to get planed to thickness. Next the template goes on it for tracing, then to the band saw with a spherical support fixture (not shown) to cut off the waste. Then back to the dome jig as in photo 37 with the template clamped in place for the flush trim pass.

Photo 38- A different tree branch after the sticking has been cut into it, and before it gets the rabbet at the dish jig.

Photo 38 shows a different piece set up on the dome after the sticking has been cut. Note that the template is no longer there, because the sticking cutter bearing rides on walnut, not the template. I made a lot of special little wooden keepers to hold parts down at each step. After each part got the sticking it went back to the dish jig for a rabbet, similar to photo 22. At this point the parts were done, except of course for making the joints on the ends.

FITTING THE BRANCHES

The first step on fitting the branches was to place each frame over the drawings I had made of the branches in the frames, and deciding the order in which the parts would be assembled. Then I marked the frame at the locations where branches intersected the frame itself. Next, I placed the first branch to be fitted on the assembly dome, and placed the frame over it at the marks. Now I traced the line of the rabbet from the frame onto the branch. This, however, was an inaccurate tracing in all cases.

Because the frame is sitting on top of the branch, the inside surface of the frame is sitting on the outside surface of the branch. But the inside surface of all parts has a different sphere radius than the outside surface. The rabbets are closer to each other because they are closer to the sphere center. So, while my scribe marks did show the correct angle to make the end cuts at, the scribe marks were too close to each other, describing a part that is too short. In all cases I had to lengthen the parts, proportionately as they were long. This is another unfortunate consequence of my decision to make spherical rather than rectilinear parts. Lucky for me I only wasted one part before I figured out what was going on.

Photo 39. Set up for coping the ends of the parts. The part must be held in place solidly, and have cope blocks on either side of it that fit tightly against it.

Photo 39 shows the initial setup for a very small branch. The branch is tilted with a spacer that you can�t see. It�s held down with a special keeper (the yellow piece behind), which was used on every single branch bar none. But, every branch had to have its own set of cope blocks custom made to accommodate the shape of the branch and the angle at which it was cut. I was able to re-use some cope blocks but rarely could they be reused without some modification, at least at the band saw. The cope blocks, as you see, have the cope profile cut into their ends so that those ends can be fitted over the sticking on either side of the branch. The cope blocks are pinched tight to the branch, then screwed down as shown.

Photo 40- The appropriate coping flush trim template is placed across the top of the pieces along the line scribed on the branch.

Photo 40 shows the flush trim template in place. It is a straight template because this end intersects a frame stile. But where branches intersected curved parts on other branches the flush trim template had to be custom made to the shape of the curve of the rabbet on the part being fitted to. I traced these, cut them on the band saw and smoothed them with a drum sander on the drill press. The accuracy of these templates was critical to the fit of these joints, and I had a few failures that had to be redone.

Photo 41- The only difference between this shot and the last is that the flush trimming has been done and now the surface is smooth.

Photo 41 shows what the piece looks like after the flush trimmer has cleaned up the end of the part.

Photo 42- After the cope. The importance of the cope blocks should be evident here. The bearing of the cope cutter needs a surface to ride on coming into and going out of the cut so that the whole cut stays properly aligned. The cope cutter bearing is not riding on the template, which is superfluous at this point. The cope block on the right must pinch the sticking to avoid tearout as the bit comes out of the cut. The cope block on the left needn't pinch, but must be close to give the bearing a surface to ride on.

Photo 42 shows the cope cut. I had the cope bit in a separate router and carriage so that I could pull the router carriage with the flush trimmer out of the template arms on the jig, place the cope carriage in, and cut. This is necessary because you are constantly alternating between flush trim and cope cuts during this fitting operation.

As each branch was completed I glued it in place before fitting others that would be attached to it. This allowed me to alter its location during glue-up slightly from what was intended, allowing me to fudge the fit and locate the piece where the joints were tightest. As well, it�s best that the parts are glued in place when you do the alignment procedure for successive pieces so you know exactly what you are aligning to.

When the branches were all glued into the frames I began a long and tedious scraping and sanding procedure on them. Looking at the door, I realized it would be best if it lay against a wider piece rather than the edges of the adjoining frames, so I made the flat spacers that fit between each of the frames. Note that these spacers intersect, three at a time, at the four top corners. Each of the three at each joint required two compound miter cuts. You might think that one compound miter cut would do all six so long as it was divided by 360o, but no. I had to make six separate compound miter jigs, a separate one for each facet. I didn�t bother trying to figure the geometry involved here, I simply used �cut and fit� until they did.

The door has no hinges. The butts of the hinges, if used, would have to stick out very far to clear the curve. So, the door simply pops off. It rests on two brass pins at the bottom and has a lock up top so that it can be secured.

GLASS BENDING

Hooray! All the parts are machined at correct radii and fitted with tight cope joints. Now all we gotta do is get some glass in it. Unfortunately, all the glass down at the local supplier is flat as Kansas, and even if it was bent it probably wouldn�t be the right curve. No two ways about it, we�re gonna have to bend our own glass.

Glass kilns are expensive but essential because you must control the conditions very carefully when the glass cools to keep it from shattering. Fortunately though, simple bending of thin sheets of glass is about the easiest procedure you can do in a glass kiln. You must make an accurate mold with a refractory cement, which is none too expensive or difficult. You must learn enough about the mechanics of heat gain and loss in glass to know that you are annealing the glass correctly (cooling). This, however, is not complicated. With these tasks under your belt, making bowls of glass at a very accurate radius is very easy. Cutting the bowls into the shapes that fit your wood will require a ring saw if the edges are irregular as in my vitrine. Ring saws are like band saws but have a 6" rigid ring instead of a flexible band. They cost $250 or so.

I researched having the glass bent by others, but finally realized that this would be very expensive because they would have to make a mold. The glass must be bent to a very accurate radius of spherical curvature, requiring such a mold. It�s easy to free-slump the glass, but inaccurate. �Slumping� is a general term for heating glass in a kiln and allowing it to �slump� down by gravity onto a mold or whatever to achieve a certain shape. �Free-slumping� is my term, for doing slumping where there is no mold.

Experimenting with free slumping to get my bowls of glass, I took metal rings and put flat glass on them in the kiln. I cooked it to where the glass began to slump into the ring, and stopped the slumping when the glass had fallen to the depth required. But the resulting bowls were not accurate sphere sections. Free-slumping cannot produce the degree of accuracy required to fit larger pieces of glass to wood, but it could work if no piece of glass must be larger than 10" diameter or so. When you slump smaller bowls to a slight drop, they approximate a sphere well (when dropped in a circle), but the larger the diameter you try, and the deeper the drop, the less accurate it becomes.

Fortunately making a mold was easy once I located the right mold material. Dan Fenton is a glass magician who lives in Oakland, California and gave me excellent advice on this and other aspects of glass work, having done this sort of thing himself for years. He suggested I use a castable refractory cement, Greencast 45L by Harbison Walker . This material is similar to concrete, but it is calcium alumina based (I think) rather than Portland cement based. It is used as an insulating material behind fire brick in furnace situations. It takes more heat than you will ever give it and it is very strong.

Beware of mold formulations you may find in glass working books. Most of these are for glass casting situations where it is essential to the procedure that the mold material be broken away from the glass casting after the cast has cooled. These mold materials must be strong enough to be cast into at high heat, but weak enough to be broken away later. You don�t want this. You want a mold that is strong as possible and that will not distort its shape in the process of making it.

Beware of making clay molds. All clays shrink as they dry and shrink again when you put them in the kiln for the first firing. They will not hold the accurate shape you need to make spherical bowls.

You can have a cast iron mold made for $750, but why bother when a few bags of castable refractory cement are less than $100? The reason would be if you needed a mold to use over and over, because a mold made of refractory cement will not hold up for as many firings as a cast iron mold. But a cement mold will hold up for the several dozen firings needed for one cabinet.

MOLD FORMING

Because the radius of curvature of the glass was different than either the dish or dome jigs, I had to make a separate dome jig for the purpose of casting the bowl-shaped mold. This wood jig is called a former, because its shape accurately forms the mold. I learned an important lesson about making accurate glass molds here. You can make a mold using a former, or you can make a mold without a former. But the latter doesn�t work so you have to use the former.

With the former made, I covered it with wax and then a 2" thick layer of Greencast, and then worked wire mesh into it for additional strength. When wet, the Greencast had an unusual quality to it, unlike Portland cement. It was rubbery, spongy, almost elastic. Quite unusual. After it set up, I pulled the mold off the former and let it sit for days before sticking it in the kiln. Before you can use such a mold you have to get rid of the excess moisture in it. If you do so too quickly in the kiln, it will explode as steam pressure builds up inside. Time is the key.

When I did put it in the kiln, I heated it very, very slowly, over almost two days to a temperature of 800 degrees (F- I use no Celcius here). I was amazed to see that even above 600 degrees moisture was being driven from the piece as condensation continued to form around the kiln lid when opened. This, I suspect, was chemically bound moisture that required the higher heat to drive it off, but I don�t know for sure.

You must use a separator on anything that hot glass will contact in a kiln. Separators are powders or pastes of high heat materials that don�t stick to glass, and prevent the glass from sticking to molds or whatever else it touches. I used a zircon-graphite paste separator used by bead makers, but discovered that the mold required very little, if it ever needed any at all. Calcium alumina is interesting stuff.

GLASS AND HEAT- SLOW MOTION

Glass is a weird substance- not a liquid or a solid, it acts a little like both. This is due to its unusual molecular structure. What you need to know is that heat moves slowly through glass, and if you let the temperature difference between one area of a piece of glass and another get too great, it will break. This is because as glass heats it expands and as it cools it shrinks, and, unlike other materials it is brittle. So, if one part of the glass is trying to expand or contract while another is not, what happens to the area between them that is getting pulled in two directions? It breaks, unlike metals which are flexible and so just bend to compensate.

The thicker the glass the worse the problem. Thin glass, like the 1/8" thick GNA (German New Antique) I used here is easiest to use. Still, you must be sure you don�t push the glass too far for its thickness, or else.

As you heat glass in a kiln, you must heat it slowly enough that the difference in temperature between the outside surface and the inside mass does not become too great. You have to let the inside catch up with the outside. Once the glass gets to a certain temperature though, it no longer matters because the glass is soft. Then you crank the kiln full bore to the temperature you need to get the glass to do the work you want. Then, once the work is done, you begin to cool it. Now you must anneal the glass, which means lowering the temperature of the glass slowly enough through a certain temperature range that severe stresses do not develop in the hardening glass. These stresses will develop if the outside surface cools and shrinks much faster than the inside mass. These stresses will be permanently frozen in the glass once it cools, if it doesn�t break to begin with, and make it harder to cut reliably. The annealing temperature range is different for different types of glass. Once you are below the annealing range you can cool the glass faster because you cannot add permanent stress to the glass anymore, but you can still break it by cooling too quickly.

I used a kiln with a digital controller, which automatically ramps the temperature up or down at a rate that you set. With thin glass you don�t need this if you know what you are doing, but until you do it�s best to use a controller, go by formulas and exceed them for safety.

TRIAL AND ERROR

Glass working is an empirical science, which means that what we know has been learned much more by trial and error than by abstract ideas. You must, however, understand the abstract ideas behind heat gain and loss with glass to a certain extent or else. Still, every different glass working operation has its own peculiarities depending upon what you are doing and how. Case-specific problems require case-specific solutions, usually depending upon the ability of the operator to understand the problem and find a way around it. As far as the glass is concerned there is no problem. It just happily bends away or shatters depending upon the situation it finds itself in. You are the one who is contriving to make it bend this way or that, and come out of it whole without excessive stress. You must arrange the situation so that the glass happily does what it will in a situation that will produce an acceptable result.

I ran into a couple of problems. On my first try, I put a piece of glass on the mold and began heating. I heated it fast because you can do that with thin glass being heated from two sides. But when I took a peek at 250 degrees, I saw that the glass was bending upward, away from the mold, and severely too! What was happening was that the mold was acting as a heat sink, so that the top of the glass was being heated far faster than the underside which faced the mold. So, it bent up as the top expanded faster than the underside. The glass was having so much fun arching its back while I was sweating buckets for fear it would shatter. The solution was to slow the rate of increase radically so the underside could catch up with the top. Once it was toward the top of the annealing range it flattened out and I could crank it from there.

Then, the glass began to slump into the mold. But, the air between the mold and glass had no way to escape. So, the highly plastic hot glass sort of bubbled up in spots rather than lie flat on the mold. It was having lots of fun lounging on a soft bed of compressed air. I was groaning the loss of expensive glass. That piece of glass was wasted, after which I drilled holes in the mold for air to escape. All subsequent pieces slumped correctly with no problem, once I let the glass do what I wanted it to do.

Dropping glass into a hole. Note that the rabbets of the wood point inward. This means that the opening of the hole on the inside is lesser than at the point where the glass resides. If you scribe the glass by the shape defined by the inner rabbet, the glass will be too small. One more nagging problem. The bigger the glass, the worse the problem. I carved some rabbets to make way for glass.

I then used a ring saw to cut out the separate pieces of glass that went in each of the 32 holes in the wood. Ring saws are glass and lapidary machines with a 6" diameter wire ring impregnated with diamond that spins in place like a band in a band saw. You can cut glass in 360o with this tool, it just grinds away at it any direction you push.

Contact me if you need more advice on glass bending, but only if you have spherically shaped wood frames to put it in!

A SIMPLER CABINET

Perhaps you want to build a simpler cabinet that uses a spherical frame. Here�s an idea.

Design a rectilinear cabinet which has only the front bowed into a spherical shape. The sides, top, and back are all flat. You will have no planes of intersecting spheres to worry about, but you will have to properly align the flat edges of the spherical frame where it contacts the flat sides and top. This is explained above in the discussion on how the frames intersect the table base of the cabinet.

Don�t use specialized router bits, just use rabbeting cutters and you will end up with a square sticking. The joints produced won�t be as strong as true cope joints but you can epoxy them.

Keep the size of the pieces of glass small so you don�t have to slump 27" diameter bowls like I did. There are plenty of ceramic kilns in the world, and if you make a mold you will probably be able to find a sympathetic potter or glass worker that will slump it for you at a fee or let you do it. Smaller kilns are not too expensive, and you will get by without a digital controller if all you do is 1/8" glass.

If you do this, don�t hesitate to contact me for advice, but ONLY if you seriously do it, not just if you dream about it, please. I�ve done enough spherical compound curve cope and stick joinery dreaming for one life time- now I want to see results.