Spanish How to determine the size, shape, and positioning of cone "skins." April 9, 2003

Question

I hope someone can help me out here. I remember seeing the formula to cut a piece of laminate, bending plywood, etc to the correct size and angle to cover a cone and get the seams correct. Next week I am going to be starting a curved wall that slopes outward at the top and could use the formula I saw and thought I would never use. The radius is 206" at the top, 202" at the bottom, 36.5" high and chord length is 106". I'm sure with the right formulas I can rout my plywood to wrap around the wall and not have a lot of hand fitting.

Forum Responses

(From WOODWEB's CNC Forum)

I think for that radius you just want an oversized piece that you trim off for the cores/bender board and then seam the laminate in. I do the math when it matters for veneer matching of tapered pieces.

This is hard to explain without a picture but I will give it a shot anyway.

To make a cone you obviously need to wrap up a Pac-man shaped sheet of whatever you are using. The unknowns are the radius and the size of the wedge that's missing. Picture a paper cup.

First the radius of the sheet is the length of the cone along the seam.

Next the circumference around the base (big end) of your cone is the distance around the sheet. What's left (this distance will be less than the circumference of the circle you've just drawn) is the wedge shape.

So for your model, it's a bit more complicated, but not much. Draw your cone and project the taper till the ends meet. That line becomes your radius for your circle. On CAD that means your cone has a radius of... (flipping to AlphaCAM here, hehe) 1891.004". I hope you have a CNC machine!

A 206" circle has a base circumference of 1294.3" so wrap that around the radius of your new circle and it takes up 39.2 degrees of the circle. So draw a wedge 39.2 degrees and that's the bit of the circle you need to use. (Since your wall is short with very little taper, the 'Pac-man wedge' part is actually bigger than the part itself - his mouth is *really* open.)

Of course since you are not making the whole cone, you can offset this outside line your wall height of 36.5 to get the proper "band" that you need.

I picture exactly what you're telling me and I even understand it! Thanks for the help.

I've done this many times in making custom trade show exhibits. The first time was a tapered turntable for BMW. It was a PITA the first time I did it but it's quite easy once you've done it a few times. The only hand fitting you should have to do is if your joints are *not* lines that span nicely down the cone, i.e. the joints in your wall skins will fit perfect unless the wall ends on both sides with that same taper, those will have to be hand fit.

Funny that you should mention BMW - this is a focal wall for a Porsche dealer. I am still a little concerned with how it is going to look when it is finished. It gets covered with brushed aluminum. Hope I get the "grain" to look decent. Personally, I think it is going to look like a mistake when it is finished.

We are a small custom kitchen manufacturer, with about 66 employees. We have a separate division, Keener Architectural Millwork, that is growing rapidly and we are doing more and more out of the ordinary things. We started out doing a lot of medical facilities and banks and now entire schools and new car showrooms.

What do you mean by chord length is 106"? I am guessing this is the length of the line formed by the intersection of the cone with the wall - but at the top or at the bottom? Also, what thickness plywood?

I am using 1/2" material and the length from point to point at the top is 106" as it is not a full round cone, only a section.

Will this work? Draw an isosceles triangle with sides 206" and base 106". Connect the base with an arc of radius 206" with center being vertex of triangle.

Slant height s =

sqrt([206-202]²+36.5²)=36-23/32"

Now draw an arc offset from the original arc by 36-23/32". Connect the ends of the strips through the center of the arcs. Erase the triangle. Is that your cone? I may have reversed the radii with the chord. Please recheck!

Here's what I came up with:

P.S. Confirm the angle mathematically with

2*arcsin(chord/dia) =

2 arcsin(106/412) = 29.81764º

I know the outer radius of the skin cannot have the same radius as any of the radii of the wall itself as it is shown above. I could be misunderstanding but I was calculating using the radius of the top "rib" at 206" and the bottom "rib" at 202". Basically your wall tapers only 4" from top to bottom, correct? That is where the huge radius comes in, as one must extend those lines quite a ways before they meet. That long line becomes your skin radius.

The chord length isn't particularly relevant here, as I don't think it is mathematically easy to figure out joints other than ones that are straight along the cone surface (perpendicular to the base "tangent" if that makes sense). I could be wrong but I've never figured out a way to get these "end panels" and they must be hand fitted. The chord length will tell you how much of the cone you will have to skin and thus approximately the number of skins you will need, cut both end skins oversize.

I just got home and had chance to reread and rethink your cone. I have it wrong.

I *think* this is the correct development of your cone. Maybe someone else will post theirs. I don't have my sci calculator and equations here, so the numbers aren't exact. I just drew it quickly on my home CAD.

Here's how I did this again. I'm (pretty) sure that 3.2 degrees is wrong.

First find the overall height of the *full* cone using CAD or similar triangles:

206"/cone height=(206-202)/36.5

(similar triangles)

cone height= 1879.75"

using a^2 + b^2 = c^2 we can get the length 'along' the cone

206^2 + 1879.75^2 = length^2

length = 1891.0 "

Draw a circle with radius of 1891"

The top of your wall has a radius of 206", thus a circumference of:

206 x pi x 2 = 1294.3"

the circle you just created has a circumference of:

1891 x pi x 2 = 11881.5"

your skin needs to have 'angled' ends (the pie shape):

angle/360=1294.3/11881.3

angle=39.2 degrees

Use a 39.2 degree 'pie' out of your circle and that is your skin. You may offset your circle for a wall that is not the full cone. In your case:

36.5^2 + 4^2 = offset^2

offset = 36.7"

I don't think that the chord length is useful in making skins other than to estimate joints or number of panels. Both ends will need to be hand fit unless the wall ends in a true cut 'along' the cone (i.e same as a panel joint, which is unlikely).

I thought this would be an interesting problem to work on. First, to get the answers for the problem, make a working example, and draw up on CAD. I wanted to see which numbers change, and which stayed the same. I got a cone shaped plastic cup, wrapped it with paper and taped it. Next, trimmed it to cup with razor. Then drew up both items on CAD.

So by looking at the relationship between a 3-d cup, and a 2-d flattened cup, I can see that the angle of the slope changes. The length of the chord would change the angle of the piece needed to laminate an object that is less than half of a flattened cone. I don't see an immediate relationship between the radius of a cone and the radius of a flattened cone.

With some investigation, I shall find some formulas to make an Excel file to solve this.

?

Looking at the drawings I made, it occurred to me that the arc length in a flattened cone is equal to half the circumference of a 3-D cone. Using that information, you can solve the problem of wrapping a partial cone with a CAD program.

If you have a chord length and the radius of a cone, you need to find the length of the arc for that.

Calculate radius times pi, divide by 2. This is half the circumference. Now subtract the length of the arc of your partial cone from the half circumference number. Use half of the answer to offset the lines of the flattened cone drawing. (The dashed lines in the drawing represent this.) This should be the piece of material that will wrap a partial cone.

View full size image

Use Autodesk Inventor 6. It will draw it the way you want and lay it out flat for you. No formulas required.

*From contributor A:*

Contributor M is correct - 3.2 degree side length is 36.7185. To lay out without CAD, do this:

1) Draw a rect. 107.1913" x 37.4634"

2) Measure .7597" down from each corner on long side on top.

3) Make an arc connecting these 2 points to center. This will be your top arc.

4) From bottom, measure in on long side 1.0407" on both ends.

5) Measure up from center of bottom long side .7449".

6) Draw an arc through these 3 points. This will give you your 102 radius for the bottom of the cone.

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