Two Cylinders Meet
If the axes of symmetry, of two 2-inch right circular cylinders intersect at right angles:
What volume do the two cylinders share?
[some of you, who have more recently visited a classroom, may visualize this intersection ... I'm having trouble seeing it ... :-) ]
What volume do the two cylinders share?
[some of you, who have more recently visited a classroom, may visualize this intersection ... I'm having trouble seeing it ... :-) ]
posted by Zaux at
6:10 AM
30 Comments:
I had to think about this a long time ... then I had to give up and Google it. I won't give it away except to say that Archimedes knew the answer, and Steinmetz named the solid formed by the intersection.
Ross ...
thanks, glad to know it's not an easy visualization ... thought I might be having a senior moment :-)
They share a 3D shape.
The volume of the shape is 5.3334 cubic inches.
Want see see it?
I have several drawings that show the construction.
First is a pair of 2 inch lines that will represent the center line of each cylinder.
Next each center line is given a 2 inch diameter circle that will the end of each cylinder. The picture is an isometric, so the circles look like ovals from this perspective. The red line is the z axis, the cylinders are in the xy plane.
Then I constructed the cylinders, one green, one yellow. This looks like a union, but it's still 2 separate objects.
One of these pictures is a hidden line picture.
Then I did an intersection that shows the 3D space that is the "overlapping" 3D space.
I made the material wood so that the wood grain helps you see the depths of the surface.
You can't see them can you?
Go to http://ragknot.blogspot.com/
Wow. I agree that was quite hard to visualise. However, that is
the hardest bit. Let the x and y axes be the centres lines of each
cylinder. Cut into two by the x-y plane. Problem now halved.
Let the z axis pass through the centre of the inersection, at 90º.
Cut through the x-z plane to make a sort of T-shape. Now cut along
the plane formed by the line x=y and the z-axis. The remaining
(small piece) looks a bit like a 1/8 slice through a square pizza
(when looking from the top. This piece is 1/16 of the volume we
want. all thos cuts, leave us with a piece that has no awkard
edges or corners, and is describe with x,y,z ≥ 0 everywhere. Nice.
The equations of the cylinders are x²+z² ≤ a² and y²+z² ≤ a².
We now need to evaluate the double integral, of z over the
domain 0 ≤ x ≤ a, 0 ≤ y ≤ x. z = √(a² - x²).
Integral of z between 0 and x (with respect to y) = x√(a² - x²).
Integral of that between 0 and a (with respect to x) is a³/3.
(Thank you wolframalpha).
The total volume is therefore 16a³/3
http://www.wolframalpha.com/input/?i=Integrate%5Bx+Sqrt%5Ba%5E2+-+x%5E2%5D%2C+%7Bx%2C+0%2C+a%7D%5D
Just noticed from Ragknot's post that they are 2" cylinders.
So a = 1 => volume = 16/3 = 5.333... cu-in.
Ragknot ...
outstanding 3-D visuals
Sadly, I can't explain the on this site (needs picture badly), how
I chose the limits of the integrals. If you have a basic knowledge
of ordinary one-dimensional integral, and would like to know more,
the best site I found (after rejecting 5) is:
http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/255doub/255doub.html
Ragknot ...
scrolled down after looking at the 3-D visuals ... saw your son's site...listened to his songs ... not bad...does he play with someone or solo? I did that for a long time before I said "I do" ... :-)
Hey, good pics Ragknot. I wish I'd had them before. It took me longer than I'd care to say, before I was sure I had good enough sketches to see the domain and function for integrating over, and to find the fairly easy to visualise shape that I ended up working with.
Would you post a picture of the 16th part of the shape, pretty please :)
Thanks for the flowers, I saw this post earlier today, but I was still at work. When I got home, I was surprised to find no solutions offered. I decided to capture some screens along the way because 3D is hard to visualize.
This is easy to do if you "get" 3D modeling. But it takes experience to "get" it. A few of my students "got" it after a few lessons, but some still had great difficulty even after a semester.
I just read Chris plea for the 1/16 piece.
OK, I will slice it out for you.
Thanks Ragknot. I've just been looking and not found a "Chris's shape". But I have found some amazing pics:
http://local.wasp.uwa.edu.au/~pbourke/geometry/cylinders/
You must keep going down the page, they end up with one made from 200 random diameter cylinders.
So although any shape can be made out of triangles, these are made from cylindrical surfaces. I'm totally stunned that they splice together so beautifully. My mind has truly be blown away by it.
Imagine a slice of honeydue melon, cut in half at the middle. But the only curved surface is cylidrical, not spherical.
Ragknot, the site listed above:
http://local.wasp.uwa.edu.au/~pbourke/geometry/cylinders
is phenomenal ... go take a look, you'll like it. Without aid of a computer, I can't imagine a human conceiving the forms.
I just posted two additional pics of a 1/16 piece. volume is 0.3333 cu in.
My son, Bill Greenway was 5 when Penny got married. He plays and sings by himself and with anyone of his friends. He plays at local clubs every weekend. He even plays for free at the lake if he does have a gig. Guitar Bill, he calls himself.
As I sort-of said in my first post ... this is known as the Steinmetz Solid. mathworld.wolfram.com has a very nice page about it here:
http://mathworld.wolfram.com/SteinmetzSolid.html
The thing that intrigued me is that the page says, but doesn't demonstrate, that the volume can be deduced without calculus. I wouldn't know where to begin...
Ragknot, thanks for those pics. The wire frame one is good. I'm going to try to see if I can imagine a sphere inside that.
For what it's worth, my domain of integration was over the only triangular surface. and I (in essence) erected vertical lines from there to the (only) curved surface. My y direction corresponds to the line that Ragknot has used to denote the curved surface.
Archimedes did some amazing stuff relating the surface area and volume of a sphere to the enclosing cylinder (and a cone).
When you look at those pics, you've got to keep reminding yourself what they are - I think these shapes I absolutely beautiful, bot visually and mathematically.
I didn't explain why I said "Integral of z between 0 and x (with respect to y) = ...)".
Looking at Ragknot's half melon slice wire-frame, I'm integrating along the line that is being used to denote the curved surface. One end of that line is on y = 0, and the other on y = x (as there is a 45º angle involved).
(This won't mean much when Ragnot pulls that drawing off; I'm sorry in advance).
Ross,
This is off the subject, but I've tried to build an html that will lead to a URL, like you did with "SOLID" in the first post. Every time I try to do that, I get a message that it's not allowed. So I must be doing it wrong.
I meant honeydew melon.
I can wrap my head around a sphere in the middle of that shape. It
will have the same radius "a" as the cylinders.
I see that the ratio of the volume of the sphere to the shape is
(16a³/3)/((4/3)π a³) = 4/π. By coincidence, that's the reciprocal
of π/4, that I recently mentioned was engraved on sliderules.
But I can't see why 4/π by geometrical reasoning.
This post has been removed by the author.
Ragknot. I took a look behind the scenes. He's got embedded
"<"A href="http://mathworld.wolfram.com/SteinmetzSolid.html"
rel=nofollow>solid"<"/A">"
I've "d the angle brackets so as not to embed it. solid that's a demo
See pictures of this Tom at... My Blog
thx - it worked
It's easy to see how to do this without calculus ;)
It is possible to fit a sphere of radius a inside the Steinmetz shape.
Make a slice, parallel to the x-y plane, through the Steinmetz shape, including the sphere. Let the slice be such that the radius of the sliced disc is r. The corresponding Steinmetz shape wil be a square of side 2r. The ratio of areas is (2r)²/(πr²) = 4/π. But this is true for all r. So the volume of the Steinmetz shape must also be 4/π larger than the contained sphere, which is (4/3)πa³.
=> volume of Steinmetz shape is 16a³/3
This sort of integral calculus was invented by Archimedes. He used similar ideas to show that the surface area of a sphere is the same as that of the enclosing cylinder. He also showed that the volume of the sphere is 2/3 volume of the containing cylinder. He had a picture of that arrangement engraved on his tombstone.
I'be made a sketch of the 1/16 of the Steinmetz Shape here: Chris's blog
This post has been removed by the author.
This post has been removed by the author.
Having just discovered Googledocs, I've published the shape there:
1/16 Steinmetz shape
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