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Math Geek Question
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ghporter
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Join Date: Apr 2001
Location: San Antonio TX USA
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Aug 10, 2019, 02:36 PM
 
I have a right circular cylinder, (circle on the x-z axes) and a plane on the x-z axes. If the plane is tilted 40 degrees from the y axis of the cylinder, is there a straightforward and direct (i.e. simple) solution to find the equation of the curve where the plane intersects the cylinder?

Obviously if the plane were at 0 degrees to the cylinder's axis, the equation would be the equation of the circular form of the circle (x squared + y squared = Constant). But I can't recall from the dim past when I took analytic geometry* how to put the problem together in the first place. In part it's because I can't recall (nor is Google helping me at the moment) how to state the plane's equation.

Any helpful suggestions, even pointing out a web page that spells out my problem in small words and short sentences, would be appreciated.

*I took "trigonometry and analytic geometry" in the first semester of my senior year in high school. That was in 1976. I have not used either trig or analytic geometry since then...

Glenn -----OTR/L, MOT, Tx
     
subego
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Aug 10, 2019, 02:48 PM
 
Far from a math geek, but I’m good at Google.

http://mathworld.wolfram.com/CylindricalSegment.html
     
subego
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Aug 10, 2019, 02:53 PM
 
If I’m understanding right, this is actually super straightforward.

The equation will be for an ellipse.

The minor axis is the diameter of the cylinder.

The major axis is the long side of a right triangle. One side of the triangle is the diameter of the cylinder. Find the length of the perpendicular side and you’re there.
     
reader50
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Aug 10, 2019, 02:59 PM
 
Assuming your plane does not intersect one of the cylinder ends, your intersection is going to be an ellipse.

Width of the ellipse will always be the width of the cylinder. Length is all you need. To get the length, look at it from the side (where your plane becomes a line). The cylinder becomes a rectangle, with a line passing through it. Then solve it as a right triangle, where the line is the hypotenuse.
Code:
__________ | |C | /| | / | | / | | / | A|--------|B | |
A to B = width of cylinder.
A angle = 40º
B angle = 90º
C angle = 50º

The rest is trig. If you want to get fancy, you might be able to simplify the equation into a general solution to the problem. Ellipse equation = the intersect angle & cylinder width plus additional terms. I'm lazy this morning, so you can do the rest.
     
ghporter  (op)
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Aug 10, 2019, 03:03 PM
 
Thanks! Of course it is an ellipse, but also of course that word was just refusing to come to the surface of my brain.

And of course you're right. That ellipse is exactly the thing. It's a 40-50-90 right triangle, which isn't one of those "do it in your head" triangles like the 45-45-90 and 30-60-90 triangles, but it's pretty doable.

Thanks for the help. I've always felt that "saying your problem out loud helps you understand it better," but sometimes the right words are hidden by overthinking the thing. Particularly for me.

Glenn -----OTR/L, MOT, Tx
     
   
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