Hi Everyone.
I am at the moment trying to veneer a sphere 800mm (2ft8in) in diameter. I am looking for a method or equation so that I can make a template of the segments to cut the veneer to shape. This problem has stumped me as I am not used to thinking in curves on three dimensions. I will need to cut the veneer flat and then mould it around the surface. Any help would be gratefully received.
Regards Peter
Replies
Jeez, I hope you math skills are alot better than mine! From this desciption of what you need to do it might be an idea to get the PhD in Geometry first;)
Date: 07/29/99 at 16:46:25
From: Doctor Rick
Subject: Re: Making spheres from paper
Hi, Matt.
I think you realize that any construction of cut and/or folded paper
will not fit perfectly over a sphere. The narrower you make your
wedges, the closer it will come, but you will never make a perfect
sphere. To be more specific, with 30-degree wedges, any cross-section
parallel to the equator will have the shape of a dodecagon rather than
a circle.
The method you describe, however, happens to be just what globe makers
do. They make pieces of map of this shape (which they call "gores")
and glue them on a sphere.
Now let's work out the shape of the gores. We'll make it so the center
line of each gore lies exactly on a sphere of radius r. Call the
angular width of each gore (the number of degrees of longitude it
covers) phi.
Look at an arbitrary cross-section of the globe parallel to the
equator. The plane of the cross-section meets the surface of the
sphere at latitude L. I will work with the co-latitude (angle measured
from the pole), in radians, which I will call theta.
theta = (90 - L) * pi/180
The radius of the circle that is the cross-section of the sphere is
r_c = r*sin(theta)
You can see this if you take another cross-section perpendicular to
this one, containing the axis of the sphere. There will be a right
triangle whose hypotenuse is r, one leg is r_c, and the opposite angle
is theta.
The cross-section of one gore is a line segment tangent to the circle
at its center. We want to find the width W of this line segment. Draw
a line from the center of the circle to the point of tangency, and
another from the center to the end of the tangent segment (the gore).
You get a right triangle with legs of length r and W/2; the angle
opposite W/2 is phi/2. Simple trigonometry gives us the width of the
gore as
W = 2*r_c*tan(phi/2)
= 2r*tan(phi/2)*sin(theta)
The distance (along the arc of the globe) from the point of the gore
to the line whose width we have just measured is r*theta. (Remember,
theta is measured in radians.) The full length of a gore, from tip to
tip, is pi*r. (The gore for a hemisphere will be half this, or
pi*r/2.) If you lay it out on graph paper so that the tip is at the
origin and the centerline is along the x-axis, then the two sides of
the gore will lie along the lines
y = +/- r*tan(phi/2)*sin(x/r)
So that's the shape of the sides of the "triangle." It's a sine curve
whose amplitude is determined by the angle phi.
As for the angle at the poles, it _will_ be exactly the angle phi.
However, the sides begin curving inward quickly, so we are talking
about the angle between the _tangents_ to two curved lines. The
equator end of the "spherical triangle" is a vertical line at
x = pi*r/2, and the sine curves are horizontal at x = pi*r/2 so the
angles are both right angles.
I haven't tried making one of these, so I don't know how easy or hard
it might be to assemble. I would try using a card stock rather than
thin paper so it's easier to handle; I would cut out the gores right
along their edges and tape them together on the inside of the
hemisphere. If that worked, I would think about fancier ways to do it.
Have fun!
- Doctor Rick, The Math Forum
I for one that read this post was quite impressed..To bad I have no idea what was said! .. Just me... If it gets outside of Electrical Engineering math I'm LOST!
At least I can follow the instructions at this site.
http://octopus.gma.org/surfing/imaging/globe.html
I think this might do it.
Lets say that you wanted each strip to be 50mm wide at the equator. If you come up 5 degrees from the equator, then you would want the strip there to be (50 x Cosine (5)) wide. At 10 degrees, then (50 x cosine (10)) wide, and so forth. So, at the top the strip would be 0 wide. (cosine(90) = 0). The table below is assuming an 800 mm diameter sphere.
At 60 degrees latitude, the circumference of the inscribed circle would be half as much as at the equator.
I have this in a spread sheet so I can run other numbers if u want.
PlaneWood by Mike_in_Katy (maker of fine sawdust!)
PlaneWood
Edited 10/2/2005 1:38 am ET by PlaneWood
Last 3 columns in above msg should be shifted left.
Edited 10/2/2005 12:58 am ET by PlaneWood
Edited 10/2/2005 1:39 am ET by PlaneWood
Thanks for your reply to my post it was the one that made the most sense.I am in experimentation mode at the moment and will let you know how things work out. I may need the equations for the spread sheet if that would be OK it will save me having to do the work again.Thanks again Peter
Dr. Ricks method may be fodder for the next edition of the "amazing adventures of Dr. Echo", but mesensed that you wouldn't have endeavoured to take on this commission or task without some sense that you would be capable of completing it, or at best if it was a commissioned piece, that you would be able to somehow pull it off.
Now, you could go the mathematical route, plotting all the required stuff and hope that it turns out as you graph then transfer the data, all the time introducing possible sources of what Dr Rick would have to call "class one errors" but what the rest of us plebes would call mistakes.
However methinks you indeed have the sphere you wish to veneer at hand. 2'8" in diameter. And if indeed you are as prudent as this request is different, you may actually have more than one such sphere at hand, ....anticipating the "just in case scenario"
Can I suggest a solution?
yer answer is too late....here it is anyways.
simply coat the sphere with a light coating of spam.
Then, perhaps in a lathe, or just simply rotated in two spindles start spraying a coating of latex polyurethane on it, which dries quckly, often times within minutes depending on heat and humidity, and should enable you to , withing a day, perhaps assisting drying with a small heater, to encapsulate your substrate with a coating of polyurethane of sufficient thickness such that you would be able to, with the aid of a suitably curved straight edge (2'8" curvature) and a really sharp x-acto or simimalr blade actually be slice off and be able to peel off the coating.
Of course, you would have to slice it at the equator first. Perhaps not cutting it right to the poles
The spam should act as a release agent and allow it to peel off easily I HOPE
That is yer pattern. No math, No miscalculations, a simple exact fit
then, you could transfer that to your veneers,
I presume you know the principles of marquetry, with angled cuts etc, I leave it to you to visualize how that may or may not assist getting good tight joints.
and there is your pattern. No fancy math, not errors, except your own, It fits your substrate exactly, and moreover, in execution becomes an exquisite epiphany of Pye's conception of the art of craftsmanship.
What I am outlining here, and am envisioning this task as a comissioned piece, ie how I would do it, and try to make money on it, is that I would have built in some costs for expirementation. So perhaps while I am cogitating how to cut the first cast into layable veneers, whilst I had the sphere already mounted to spray, I would be spraying the second cast, just in case I screwed up the first.
And anyone who has tackled such a wierd (these days) project would likely not disagreee with me that it's not unheard of to f*ck it up the first go at it. Prudence and risk management ain't foreign concerns here.
After all, you got at least 20 to 30 curved joints to pattern and transfer on each side of the equaltor without flaw or defect, perhaps to expensive and delicate veneers.
Then, maybe even generating a third or forth latex poly pattern is worth considering, in the terms or "risk management" or more pragmatically, economic utility
IIs it concievable that you have pushed yer limit here buddy, and if so, more power to ya, To accept a commission to do such a task without conception of facilitation of same takes a few cajones, or even to push the envelope of capability for yourself without any clear perception of how to complete it means "casting your fate to the winds" , and that aint the act of an intrinsically shy person. Hey, maybe you read the first edition of Dr. Echo's book.
And I will tell you that I ain't never tried this exact task myself, and any advice I give ya has gotta be considered as naive advice, but on the other hand, the problem puzzled me and I could not let it slide without adding my two bits.
Remember in high school when they taught you "geometry", well that was planar geometry, They also taught you triginometry, and that was "planar trig" too, too bad they never told you (or me either) that the earth was round, so all this planar crap was just the tip of the iceberg, the world is round, ie spherical, and that there were concepts like "spherical geometry", which is so tied to triginometery that it defies logic as to why the teachers didn't make the connection visible to their students.
So any navigation, tall builiding construction, surveying, well that's all "spherical geometry" and it's obviously what navigates aircraft, applies to surveying (ie yer property lines) and all the rest of what the Flat Earth Society denies. Don't for a minute think that you need a Ph. D for what Dr. Rick espouses. A simple trip to the library to obtain a CRC handbood of mathematical equations will within 5 minutes reveal more capability of maniputlation of mathematical data than 4 years of high school will teach ya (provided you learned to read somewhere along the way) Hey , all you gotta do is start reading the index and you'll be shortly flabbergasted as to what you can do with a little math knowledge, and what little you were taught, although methinks yer not exactly naive in this area. Certainly no insult in intended in those statements. The mere fact you are asking this question places you well in the sphere of human endeavours, at least to my mind.
Now as for glueing, methinks a vacuum method might be best, but I've got no experience in spherical vacuum glueing, any others feel free to chip right in here!!! but yer latex polyurethane pattern could/ would also be the pattern for a sphere, or else you could find a beach ball of appropriate diameter and simplye suck it down instead of blowing it up. exquistiely simple, but you might look just a tad wierd measuring beach ball diameters in the stores......
My advice was free except for one caveat....you gotta let me know (pictures if possible eh?) how it turned out and if my observations and suggestions were of any benifet to ya.
An interesting post, big fella
Eric in Calgary
Eric, your beach ball idea is interesting, but how would you get the globe into the bag?
Steve
Scissors, and patch it up with tape. the trick would be to find the right sized beachball...
I thought about this a few times today. Maybe the polyurethane varnish spray might be sprayed on something like wet tissue paper to give it more body.
What an interesting puzzle.
Eric
Hello Eric. I was very interested to read your post to my problem. I am making this as a spec piece and for the challange. As yet I haven't made the carcase and my end up makeing that out of veneer. Thank you for your advice and I will reply when I have completed the project. It may be a while.Regards Peter
" A light coat of SPAM" Do you mean PAM? Or do you mean to rub a hunk of Spam on the globe? Sounds like something out of Monty Python. Aside from the latex poly there are products on the market specifically for this purpose. They are avilible at craft and hobby stores.
Mike
Lots of good info on the replies here so far. You might also want to head to a Woodcraft store and browse "CIRCULAR WORK IN CARPENTRY AND JOINERY" by George Collings. I borrowed the book from a neighbor once. There's tons of good info in it and I'm pretty sure the section on building domes has info that would be applicable to what you're trying to do.
Waddaya mean it wont fit through the door?
Peter, here is how I'd go about it. I've never veneered a sphere before, and it might not work, but I think it would.
First, forget about the formulas and templates. In my experience with veneering, if you cut something to exact specs to fit a hole, it won't fit. It'll be too small, which is hard to fix. Too big in no problem though, so on that pretence, here would be my plan of attack. I'm assuming you know the basics of veneering, so I won't include the basic stuff.
Draw lines around the sphere, just like you have for a globe, an equator, and the lines passing through the poles (longitude or latitude, I forget which) so that you get the number of fields that you think looks like it will be a good tradeoff between flat pieces of veneer and a managable number of pieces. Now get a bottle of Super Soft 2 veneer softener (diethylene glycol monoethyl ether) and treat your veneer so its nice and pliable. With this stuff you can wrap a burl around a pencil. You'll need a vacuum press. I'd start with a couple of those fields on your globe at a time. Lay the softened veneer on the globe, and cut it close to size. Leave it a little big if anything. Now glue the field, stick the veneer and press. After the press, you can trim the stuck veneer to fit those lines, and scrape off the glue that is where you don't want it. Repeat about a million times, and you'll to the last field, where you will be able to manipulate the size as needed. Because you're dealing with wood, it'll press and flaten inconsistantly, so you will have discrepencies. That is why I'd use this method rather then the mathmatical approach.
Whew. What you are trying to do is an approximation that has stymied map makers for over 500 years. Namely, how to present a map of the world (a sphere) on a flat 2D map.
Note the word "approximation." There is no exact transformation from a 3D sphere to a 2D representation. All maps have some distortion in them; translated to your problem, your veneer will not fit exactly to a sphere no matter how you cut it. The best you can hope for is something close enough where hammering in edges will get you a decent seam.
There are many approaches to this problem that have been used by those with mathematical PhDs. Take a look attheir work by google'ing world map projections.
Good luck!
I think if I were to try this, I'd consider not veneering at all. That is, GLUE UP your globe from a series of flat pieces, sort of like you can turn a cylinder from a series of pieces glued into octagons or whatever. Turn your globe from that.
Of course, if you're trying to do accurate shapes, like coastlines, that would probably be nearly impossible. But if your "globe" isn't a recognizable likeness of the earth, maybe you can do the design you want this way.
16 posts and still going? It is impossible to do in any way that would be called suitable or better, despite the math and oh so hopeful conjecture. The mapmaking history comes closest to closing the door on the operation. A simple visualization or thought experiment will indicate the impossibility.
I wonder what this thing is and why wood? I thought globe right off the bat. And Fine Woodworking - back in the days when it was a resource for methodology instead of filler for tool ads - had a great in depth article about building a large hollow sphere out of solids - a globe. It was done with curved parts mounted on a pivot jig and sliced on a table saw. Very cool. The sphere was built up in two halves then joined. Smoothed out by hand and then carved with the continents, etc.
Not really the same problem as what you're trying to do, but I remembered this related post from a while ago and thought you might be interested.
http://forums.taunton.com/tp-knots/messages?msg=23152.1
Waddaya mean it wont fit through the door?
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