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Cylindrical Surface!

  1. Apr 19, 2004 #1
    What is the equation of a cylinder with its axis in the xy-plane and making an angle 'alpha' with the x-axis, the axis intersects the y-axis at a distance of 'k'?
    Initially i thought this problem to be very simple but haven't got any success with it in last few days

    thanks for your help!
    Xishan
     
  2. jcsd
  3. Apr 19, 2004 #2

    Integral

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    Take the expression for a cylinder aligned with the axis, apply a rotation and translation of your coordinate system.
    For a translation
    [tex] x = x' + h[/tex]
    [tex] y= y'+k [/tex]

    for the rotation
    [tex] x= x'\cos( \theta) + y'\sin( \theta) [/tex]
    [tex] y=x'\sin(\theta)+y'\cos(\theta)[/tex]
     
    Last edited: Apr 19, 2004
  4. Apr 20, 2004 #3
    No sir!

    When the cylinder's axis lies in xy plane and is NOT PARALLEL to any of the axes, shouldn't the equation comprise of all the coordintes (i.e., x, y & z)?

    What you've given here is OK for an in-plane rotation or translation but not for my case! or is it? This way the cylinder is rotated about its own axis which for a right circular cylinder doesn't need any axes transformation at all!
     
    Last edited: Apr 20, 2004
  5. Apr 20, 2004 #4
    Xishan,

    Your original question said the axis is in the xy plane, but not parallel to x or y. Integral's rotation will make it lie along the new x (or new y, I can never tell which until I've done the rotation!) axis.
     
  6. Apr 21, 2004 #5
    I've just managed to solve the problem, the equation of that cylindrical surface turns out to be,

    x^2 + y^2 sin(a)^2 + z^2 cos(a)^2 -yz sin(2a) <= r^2

    this cylinder has its axis in the yz plane and makes an angle 'a' with the y axis in the ccw direction. This can now be verified: putting a=0 gives the equation of a cylinder with its axis along y axis,
    x^2 + z^2 <= r^2

    and for a = 90,
    x^2 + y^2 = r^2, a cylinder with its axis along z axis!

    now if the axis is moved away from the origin, the respective intercepts may be subtracted from x, y or z.

    Thanks everyone for considering this problem!
     
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