cylindrical geometry question

May 2010
8
3
Hi,

Here is my challenge to you:

I have a cylinder with its axis of rotation on the x axis and of radius r. I cut the cylinder with two planes both perpenticular to the y-x plane and passing through the origin: one at 45 degree from the x axis and one at 135 degree. I understand that I will get two elipses on the surfaces of the cut cylinder but my question if I "peel" the cut cylinder (only one piece) as if it was made of fabric and flatten it, how can I draw this piece?

How could I get the distance between a and b as a function of x? or just the equations of the two curves?

any idea would be aprecieated.


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May 2010
27
5
The center height of the peel would be half the cylinder circumference.
The maximum height of the curves at each side of the center height, would be the same as the cylinders radius.

hmmm, I don't think the curves map to a scaled circle? no, not since the 'edges' will have a cut at the tip of 45+45 degrees and a scaled circle would have a continuos edge.

//Edit:
I wonder if not the curve maps on a graph from 45 degrees to 0 degrees as you hit the maximum curve height?
so that would be the derivative(?) of the function as y peaks at 1/4 of the circumference of the cylinder? (I rotated your diagram 90 degrees)
I'm sorry I'm totally algebraically illiterate... but yes, that is a solution.
 
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May 2010
8
3
great! thanks, that helps alot!

I dont know algebra very well either, I just didn't even think of thinking about it analytically...
 
May 2010
27
5
my brain couldn't quit so I made a formula anyway.
I think it is correct, I just I never thought of it as a straight 45 degree cut through a cylinder...

if d is your circumference/4 of the cylinder
and d=x in the x,y system.
r = radius
...and you use radians, the formula should look like this (I think)

\(\displaystyle y =r \times \cos \left( \frac{x}{d} \times \frac{\pi}{2} \right)\)

and you plot from -d to +d.
 

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May 2010
8
3
Thanks!
I am in the middle of finals so I was going to work on that next week but you did all the work for me :)

It was a fun problem though,

Thanks again,
 
May 2010
27
5
aah, sorry, how stupid of the, it is the end of the semester now.

Ask your teacher for a new problem, I can hardly imagine that would hurt your grade. Was this problem specifically for you? In that case your teacher likes you. Go ask for another problem :)