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Math Help - Intersection of cylinder and plane

  1. #1
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    Intersection of cylinder and plane

    "Let S denote the elliptical cylinder given by the equation 4y2+ z2=4, and let C be the curve obtained by intersecting S with the plane y=x.

    Parameterize C


    I am not sure how to go about this. I tried solving the equations for each other

    4y2= 4-z
    y2= 1- z2/4

    y= sqrt(1-z2/4)

    and then do i do some sort of parameterization of sqrt(1-z2/4) -x=0 or am i supposed to parameterize the cylinder and plane separately? I know the cylinder parameterizes to
    y=cos(theta)
    z=2sin(theta)
    but i don't know what to do from here. If anyone could hit me with some tips i would greatly appreciate it



    EDIT:

    would it simply be <cos (t), cos (t), 2 sin (t)> as x=y and y is equal to cos(t)? is it really this simple?
    Last edited by bchemmath; February 18th 2013 at 11:23 PM.
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    Re: Intersection of cylinder and plane

    Quote Originally Posted by bchemmath View Post
    would it simply be <cos (t), cos (t), 2 sin (t)> as x=y and y is equal to cos(t)? is it really this simple?
    Yes, that's correct.

    - Hollywood
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    Re: Intersection of cylinder and plane

    The next part says:
    " Use the parametrization above to compute the unit tangent vector, the principal normal vector, and the binormal vector at each of the two points where C intersects the xy-plane"

    I know how to solve for unit tangent vector/ principal normal vector/binormal vector, etc but it says at each of the two points where C intersects the xy plane. Doesn't C intersect it in its entire domain? Should i have parameterized this differently without using sin/cos? or is it just at 0 and 2pi since that is the edge of the domain? i am slightly confused if anyone could please explain to me
    Last edited by bchemmath; February 19th 2013 at 07:59 PM.
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  4. #4
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    Re: Intersection of cylinder and plane

    The xy plane is where z is zero. Since in your parameterization of the curve, z=2\sin{t}, it crosses the xy plane where \sin{t}=0, i.e. at t=0 and t=\pi.

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