Bartok's Theorum?

[Tom R]

Although this is going to be a tough one (that's why I'm 'pickin' on Joe), - - it's actually a 'real-life' construction math problem. The object is to cut 'dadoes' in my tapered kitchen architrave columns to recieve 'flush' laminate feature strips that match my existing laminate countertops and breakfast bar. I'll be 'recessing' these feature strips at both the tops and the bottoms of each column, - - but it's only the tops that are tapered. First is a picture of the kitchen columns and then a few (approximate) dimensions/drawings. I'm mainly interested in a 'working' formula, - - then I'll plug in the actual numbers when I'm ready to start 'wreaking' the havoc! (Don't you just love 'high-risk' operations on expensive materials?). I'll also be 'fluting' these columns but that's just a matter of making a 'box-jig' (as if I have time for all this).

Anyway, Joe, - - if you can't come up with the 'formula', scanning a CLUMP-OF-YOUR-HAIR will 'suffice'! :twisted: Good Luck! - - I've got faith in you!

[Tom R]

I think it's 80" for the inside diameter, and therefore 81" for the outside diameter, - - but I only arrived at those answers with my 'barnyard' logic, - - I'd still like to know an actual formula. Here's my 'cheater' method - -

[Tom R]

To cross-check if I took the known circumferences, - - and divided the 20 into 20.25, - - I come up with 1.0125

Then if I took the circumferences of the 80" and 81" radii, - - and divided the 502.625 into 508.9375, - - I come up with 1.012559

I'm basing these ratios on the theory of circles and segments of circles all being directly proportional to 'Pie'.

If this is correct I guess I can live with bein' off by a half of a ten-thousandth of an inch for now.

If I'm as right as it seems, - - maybe I'm better off just stayin' in the barnyard! :D

[Tom R]

Thinking about it more this morning, - - if I took the 1/4" difference in the two circumferences, - - and multiplied by 4 to make the 1" radial difference, - - and therefore multiplied the 20" and the 20 1/4" by 4, - - I'd come up with the 80" and 81" radii (it can't be that easy, can it?) :roll:

[Rich]

sitting here scratching my head

[Joe Bartok]

I couldn't help but notice that this problem is phrased in my simple, easy-to-understand style. :twisted:
Just kidding! Tom, is there any chance you can post your drawings in something like word, or a pdf file? I can barely make out the sketches, and can't see the numbers at all. I tried downloading the sketches and enlarging them, but I don't have a decent program that will enhance images (just MS Paint, and it can't get information out of the jpgs that isn't there to begin with). Sketches are fine, I don't need AutoCAD quality drawings, as long as the information is visible.

[giddonah]

R=C/(2pi)

R1=20/6.283 =3.183" This is the radius for the top part
R2=20.25/6.283=3.222" for the bottom part

If you draw those two circles with 1" beween them and 20.25" on the outside (measured with a string or seamstress tape) and 20" on the inside, it should work. Give me a sec, I'll draw a picture.

[giddonah]

I can't add attachments?

:evil: it's only 17k :(

[Joe Bartok]

Tom: I'm not sure what kind of a "formula" you're looking for, I went over your numbers, and they seem fine. Don't knock that "barnyard logic", it usually works fine. Radii and diameters of circles are related to their circumferences in direct proportion by pi. It is as simple as that.
By the way, don't worry about "picking" on me. I found this website by chance, and registered on the spur-of-the-moment to answer a question, but I'm still hanging around because the people here are going to have questions to problems I've never thought of. Some of them may even stump me. But no matter what, I'm going to get a chance to hone my skills.
Do post some bigger pics if you can. I'd like to make sure I'm not missing something. Looks like I won't have to post any scans of my hair. Not yet, anyway! :)

[Tom R]

Sorry about the 'dim' sketches Joe (think I need a new print cartridge). What little computer knowledge I have is already 'quite the stretch', - - I don't know how to do it any other way.

But basically you can just picture a long, narrow, upright cone, - - with such an acute angle that the 'plane' of it's 20" circumference point is located exactly 1" above the plane of it's 20 1/4" circumference point.

The problem being to find the 'inner' and 'outer' radii of that 1" strip when it's cut out and removed from the cone and laid on a flat surface.

In other words, - - two segments of a circle located exactly 1" apart, - - the inner segment is the 20", - - and the outer segment is the 20 1/4", - - and we need to determine the radii of each, - - the inner radius would be 1" less than the outer radius.

My 'figurings' seem to be correct, - - I'll have to try and think of something else to stump you! :twisted:

I have no problem confusing myself, - - looks like YOU are going to be a little tougher!

Although it does sound like Rich might owe us a little HAIR-SCAN! :D :twisted: :D

[Rich]

Well your last post filled in the blanks for me.
I was trying to see your difficulty - but now by laying it flat I can see it. I'll see if I can CAD something up. Maybe from that we can come up with a formula.

[Joe Bartok]

We need a development of the cone. I'm working on it, but my resources in terms of a drawing program are limited to MS Word. Maybe it's best to wait for the CAD drawing.

[Tom R]

Giddonah, - - I don't think you're understanding the whole problem, - - (meaning you get just one more chance before 'owing' us a HAIR-SCAN! :twisted: )

I'm actually after the 'radius' of those two numbers AFTER they become just 'segments' of a much larger circle, - - that are located exactly 1" apart.

In other words, - - getting back to the 'real-life' problem at hand, - - it I was to simply 'straight-cut' a 1" wide strip of laminate, - - and try to wrap it around a 'tapered' column, - - the bottom of the 1" high strip would be 'tight' to the column, - - but the top would have a gap all the way around, - - therefore the 1" wide strip would have to be cut at a very slight radius to sit flat on the taper.

Those two radii, - - it turns out are 80" and 81", - - so what I'll do (when and If I can ever get to it), - - is cut a 1" wide piece of wood to those radii, - - and then use it as a pattern to 'rout' the laminate.

P.S. My wife says it reasons like this (that I keep 'inventing'), - - that this kitchen will never be finished!

[giddonah]

It would all have been clear long ago if I could get an attachment on this thing to work :?

No, the radii difference is not 1". I made a drawing that is clear as day, but alas...

[giddonah]

If the radius were 80", you'd be looking at a circle of 502" circumfrence.

Blast. Someone give me an email address I can send this attachment to.

[giddonah]

I just printed it out, cut it, and put the two ends of it together. It works perfectly.

[Tom R]

Damn, Rich, - - I thought I had ya'!! :D

It's always 'tougher' to picture what 'someone else' is talking about. :wink:

[Rich]

It's at the bottom of all my posts :)

[giddonah]

That won't let me attach anything. I need to send it via hotmail or gmail.

[Tom R]

Correct, giddonah, - - the circumferences are 502.625" for the 20" segment, - - 508.9375" for the 20.25" segment, - - therefore the 80" and 81" radii are what would be necessary to make the 1" strip lie 'flat' (on this particular taper).

[giddonah]

Ok, well, your first drawing says 20" and 20.25" circs. and I went off that. It's still the same shape though, just use 80" and 81". the trick is to make sure the arc lengths are right on the inside/outside of the conic section.

[Joe Bartok]

Check this out. This is the cone in section, NOT TO SCALE.
Solved using trig, and the figures concur with everything posted so far, regardless of the method used. Now to see if I can sketch a decent development.
Am I understanding the problem?

[Tom R]

As tough as this problem 'seemed' when I first thought of it, - - it's actually so easy it practically slaps you in the face, - - just a matter of 'ratio-multiplication', - - based on simply 'attaining' the exact radial difference between the two 'given' segment lengths. 8)

[Joe Bartok]

Yes! That's what you've been doing all along. I'm still trying to make up a decent sketch of the developed cone.

[Tom R]

Yup, - - you got it, Joe, - - now if we can just figure out why our answers 'differ' by that last little 1/20,000th of an inch :D , - - I'm guessing it's the 'calculator' at fault (can't be me, right?) :D

[Tom R]

I imagine it has to do with never getting to that last piece of the 'Pie'!!

[Joe Bartok]

The answers differ due to rounding off of the calculations. Trig functions, like the number pi (which they're related to), are like the "Energizer Rabbit". They keep going ... and going ... and going ... I don't usually worry about that 1/20000th, since none of the $#!tty saws on the market will cut that close.
One thing does bug me still: I'm thinking we need the radii on the nappe or surface of the cone. Here's what I get using trig (or just use the Pythagorean Theorem):
3.222888 / sin 2.27854° = 81.06356
3.183099 / sin 2.27854° = 80.06277
This is about 1/16th longer than the actual height of the cone.
Back to the development; I don't know if I'll get it done today!

[giddonah]

Beaten to the punch by someone who can do attachments.

When you unfold the cone, the "flat version" radius is the distance from the top circle (the 3.183) to the vertex.

sqrt(3.183^2 + 80.999^2) for the inside radius and that +1 for the outside. Then cut out of that circle the arc who's inside length is the circumfrence of the top part and outside is the length is the circumfrence of the bottom part.

[Tom R]

I think maybe the calculator has 'Pie' rounded 'up' to the next slice! :D

[Joe Bartok]

giddonah, you suggested one of the pdf creators!
Anyway, I'm going to hold off on more drawings. Looks like all the bases are covered.
I had a problem when uploading my first jpg, which also was well within the file size and type parameters. The first two upload attempts were taking forever. So, I gave up, and tried later. This is the strange part: I had saved this jpg, it had a file name, a size, and when the mouse hovered over it, there was a preview. When I went back to try the third upload, I actually opened the file, and there was NOTHING in there. Figure that one out! I never did figure out what happened. (And this information probably doesn't help you much either). Well, this was fun, fun, fun. Anymore questions?

[giddonah]

The problem isn't with creating the image. I have acrobat pro infront of me, photoshop, coreldraw, powerpoint and illustrator. I just didn't notice the allowed extensions thing, I just looked at the size and figured it would take it. :oops: It's worthless now, the numbers are wrong.

[Tom R]

That's all-right, giddonah, - - thanks for the input, - - thought I'd get at least 'someone' to pull some hair out (besides me), - - oh, well, - - maybe next time, - - gid-darn-it!! :lol:

[giddonah]

gid-darn-it!! :lol:
hahahahha

[Joe Bartok]

Now that this masterpiece is done, it's probably a waste of posting it. The piece of the "pie", or "pi". I sure hope it's the "thought" that counts, or at least the numbers, and no one is scaling anything off these drawings.
The mention of attachments brings to mind a question. And Tom, I don't think it's your printer that's at fault with your images. It's the size, and when I last posted a jpg, the result was the same, the image was small, and details were hard to make out. In the "Show Off you latest Creation" gallery, the images are full screen size. Am I doing something wrong?
... ... ...
Edited Jan. 08 / 2005
I thought about the dreaded cone some more overnight, and had time to come up with another masterpiece. Yesterday’s development of a cone was kind of half-a$$ed, so it’s been deleted and replaced with this better version. (And hopefully correct version; no Internet last night, so this is done from memory). The graphics in MS Word are limited and this is still by no means close to scale, but the illustrations demonstrate what’s happening without a lot of calculation. The process can be used to develop any cone, so this could be considered a formula of sorts.
A couple of points of interest: Note that yesterday’s 1.0125 proportion is still valid. In another post, I remarked that an angle is actually a ratio, measured in radians, and equal to the arc length / radius. This is exactly what’s happening in the calculation for the sector angle, it’s a natural part of the process, and some textbooks actually refer to radians as the “natural” system of angle measurement. The factor 180 / pi has to be introduced to convert the result to the more familiar degrees.
Anyway, Tom, your figures were correct from the get-go, as were the numbers everyone else came up with. And here’s your procedure or formula for future reference:
(1) Beginning with the height of the cone and the radius of the base, use the Pythagorean Theorem to find the length along the surface of the cone. This is the radius of the circle for the development the cone.
(2) Radius of Base / Length along Surface = Sector Angle (the diagram uses the circumferences, which are proportional to each radius by 2*pi). Multiply this value by 180 / pi and lay out the sector angle.

[Tom R]

That definitely puts it in a more 'exacting' perspective, - - good stuff Joe!! :wink: