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Parametrizing Intersection of Cylinder and Plane

  1. Apr 10, 2012 #1
    1. The problem statement, all variables and given/known data
    Let C be a cylinder of radius 1. It is cut by the x-y plane from below, and by the plane z-x=1 above. Parametrize all the surfaces of the cylinder. Find a unit normal (pointing outward) for each surface.
    2. Relevant equations
    Equation for a cylinder:
    [tex]x^2+y^2=1[/tex]
    Equation of the plane:
    [tex]z-x=1[/tex]
    3. The attempt at a solution
    Bottom:
    [tex]g(r,\theta)=(rcos(\theta),rsin(\theta), 0)[/tex]
    [tex]n=(0,0,-1)[/tex]
    Side:
    I'm not so certain about this one... it should be a function of the height as well as the angle, but I'm not certain how to restrict the angle to depend on the height...
    I guess something like [tex]x^2+y^2 \le z-x[/tex]
    I think this is the normal, though...
    [tex]n=(x,y,0)[/tex]
    Top:
    I should compute the intersection of the plane and the cylinder. So I get
    [tex]x^2+y^2=z-x[/tex]
    [tex]z=(x-1/2)^2+y^2-1/4[/tex]
    And what now?
     
    Last edited: Apr 10, 2012
  2. jcsd
  3. Apr 10, 2012 #2

    LCKurtz

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    Looks good for the bottom.

    Your original thought, height and angle are the natural parameters. So try using ##z## and ##\theta## as your parameters.
    The top is the slanted plane ##z=1+x##, not the "intersection of the plane and cylinder". You might try ##x## and ##y## as the parameters.
     
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