Geometry question about planes and cylinders.May 2, 2011 2:42 PM   Subscribe

Could someone help me with this geometry problem? Cylinder with a plane bisecting it at a 45° angle...

I need help with this - I could just play with it in cardboard until I got it right, but if I knew the math it would make it tons easier.

I am making a bolster pillow that is an 8 inch diameter cylinder. This cylinder will have two 90° bends in it.

How do I figure out the shape it would be flat? I know that the shape created when it is sliced through is elliptical and I know how to draw an ellipse, but need to know how to draft it if I am going to then slit the cylinder and open it up. Is the top edge going to be a parabola?

I am looking at this.

I am looking at the second from the top example - I know the dimensions I need for the entire piece at the inside and outside measurements, but need to know what it would look like flat so that I can lay it out and pattern it.

Links to instructions and the formula?

Thanks!
posted by Tchad to Science & Nature (19 answers total) 2 users marked this as a favorite

Use string.
posted by jon1270 at 2:46 PM on May 2, 2011

And if you haven't figured out the longer dimension, it will be the diameter * the square root of 2.
posted by jon1270 at 2:48 PM on May 2, 2011

This is mental geometry, so I could be mistaken, but...

I believe the top of the sheet will look like one period of a sine wave: trough to peak to trough. (So actually negative cosine.) The amplitude of the wave will determine the angle. Overall amplitude will equal the difference in height between the two sides of the ellipse. You want that to equal the diameter of your cylinder.
posted by supercres at 2:55 PM on May 2, 2011

...so if that's right (again, big if), the period of the cosine wave would have to be equal to the circumference of the cylinder. (Trough to peak times pi.)

So if you want a cylinder two units in diameter, the equation for the top cut would be y= -cos(x)

I think that's right, but I don't trust myself entirely if I can't print it and try it.
posted by supercres at 3:08 PM on May 2, 2011

Supercres has got it: if you cut a cylinder at an angle and then unroll the cylinder, the top edge will form a sine wave. I'd still try constructing a model first to make sure that you get the seam along the cylinder where you want it to be (supercres's model runs the same into the "crook of the elbow", I think), but at least there will be less trial & error involved.
posted by Johnny Assay at 3:16 PM on May 2, 2011

I will be trying to work this out for real in about an hour - thanks, guys! I think that Supercres is right, but it has been a long time. I will come back after 9 and let you know what happened.
posted by Tchad at 3:17 PM on May 2, 2011

A parabola and half of a sine wave are not the same as half of an ellipse. A sine wave or parabola do not meet the axis perpendicularly. See in this diagram how the top half of the ellipse meets the A-B line perpendicularly, and in this diagram how the parabola does not meet the x-axis perpendicularly...and here is a sine wave.

Now: to calculate the ellipse, you have the short diameter already, 8 inches=the diameter of your circle. To get the long edge, you need to know that you are making a 90 degree bend, which makes (in profile) a right isosoles triangle, the long edge of which (hypotenuse) is equal to the short edge times the square root of 2.

So your short diameter is 8, your long diameter is 11.31 inches. Depending how accurate you need it to be, you can rough it in from there, or calculate by hand and use the string method as jon1270 suggested.
posted by lemonade at 3:27 PM on May 2, 2011

Sorry, I meant you can calculate [more specifically] the foci and then use the string method.
posted by lemonade at 3:31 PM on May 2, 2011

I suggest a very different approach to making your pillow.

Simply lay out a rectangle of cloth with a length equal to the total outside length of your desired pillow plus twice the extra length you'll need for the closure at of the cylinder at either end, and width equal to the circumference of your pillow plus the extra width for the seam.

Sew the cloth up into a long. straight cylinder. At that point the inside of the finished pillow will be on the outside.

Fill the inside out cylinder loosely with fill and bend it roughly into the shape you want. Then go to the bends and pull (or push from the inside) the cloth at the inside of the bends out and pinch it together so that there is no fill between the pinched together pieces and there are no wrinkles left on the inner and outer surfaces. Tack or clip the base of the pinches together, and sew through the line of the tacks.

Empty out the fill and turn the pinched and sewn cylinder inside out. All the seams will now be on the inside. Put the fill back in and voila! You have a pillow with two 90 degree bends with two invisible flaps inside and two sewn seams at the bends which form part of an ellipse, approximately, that the outer surface of the pillow at the bends completes.

And you haven't had to cut anything except the initial rectangle of cloth.
posted by jamjam at 4:02 PM on May 2, 2011 [2 favorites]

Hmm, in contradistinction to the way I was initially visualizing it, the seam at the base of each pinch would need to be curved in the opposite direction of the outer edge of the pinch, wouldn't it?
posted by jamjam at 4:51 PM on May 2, 2011

This is how I would figure out the shapes: Take a cardboard paper towel or toilet paper spool, slice it in half at a 45º angle. Rotate those pieces and fit them together at a 90º angle just to check, then cut on your seam lines. scale up to the size you need, including seam allowance. Then you have patterns.
posted by oneirodynia at 5:05 PM on May 2, 2011

I sliced a cylinder at 45 degrees and unrolled it in a CAD program.
Unrolled sliced cylinder
If you scale that shape so that the horizontal length is the circumference of your cylinder (~25.13" for an 8" diameter), that should be the pattern you need. Of course the height needs to change according to the length of your cylinder.
posted by hot soup at 5:16 PM on May 2, 2011

How about this: Make a straight bolster in your desired diameter, use string (as jon1270 says) or a rubber band at a 45 degree angle around the bolster, then draw the path of the band on the bolster. Take apart the bolster and flatten it , and you'll have a template of the pattern you want in the size you want it.
posted by ShooBoo at 6:44 PM on May 2, 2011

A parabola and half of a sine wave are not the same as half of an ellipse. A sine wave or parabola do not meet the axis perpendicularly.

No, but when you roll it up the edge of the cloth will form an ellipse. We're saying that the angled ellipse that forms the end of the tube will, once you unroll it, become a sine wave. You can see from hot soup's diagram that the lines don't meet the axis perpendicularly.
posted by Johnny Assay at 8:36 PM on May 2, 2011

No, but when you roll it up the edge of the cloth will form an ellipse. We're saying that the angled ellipse that forms the end of the tube will, once you unroll it, become a sine wave.

Ah, you're quite right--I read the question as asking how to find the size of that sliced ellipse, not the actual question of what does the sliced tube look like.
posted by lemonade at 9:53 PM on May 2, 2011

Sorry for my useless answer at the top. I was thinking like a woodworker instead of a seamstress.

You could plot out several points with a little bit of trig and then connect the dots, but scaling Hot Soup's CAD drawing is probably easiest.
posted by jon1270 at 3:41 AM on May 3, 2011

Wow. Thanks, everybody!

We didn't get to it last night - Monday night's classes (this is for an adult sewing class) are really chatty and they don't get as much done as they could.

I am going to shoot for the mathematical answer first. I am always telling them that basic geometry comes in handy for all kinds of sewing and design, but we usually just touch on triangles, circles, and ellipses. This should be good for them.

If we can't get the trig right in an hour or so next Monday, then I will just pull up Hot Soup's rendering and go from there.

Thanks again!
posted by Tchad at 10:47 AM on May 3, 2011

If you start with y=cos(x), the period is 2*pi, the amplitude is 1 (from center to peak).

Your construction cutting a cylinder with radius=4 on a 45 degree angle requires a period of 8*pi and an amplitude of 4. The equation for that cosine wave would be:

y=4*cos[x*(2*pi)/(8*pi)]

or simplifying:

y=4*cos(x/4)
posted by huckit at 11:52 AM on May 3, 2011 [1 favorite]

Just to show work:

The cylinder is defined by the equation x2+y2 = 16, and the plane z = x crosses it at a 45° angle. We can therefore think of the ellipse as the set of all points of the form (4⋅cos θ, 4⋅sin θ, 4⋅cos θ).

If we unwrap the cylinder, we need to express θ in terms of the length L of an arc on the cylinder. Since the radius is 4, L = 4⋅θ, or equivalently θ=L/4.

This gives us z = 4⋅cos(L/4), or what huckit already said.
posted by Elementary Penguin at 4:58 AM on May 4, 2011

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