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Parameterization of hyperbola intersecting cone

  1. Jun 8, 2009 #1
    Hello. I am having some trouble with the following problem and would be thankful if any of you could help me out.

    1. The problem statement, all variables and given/known data

    Let C be the hyperbola formed by intersecting the cone

    [tex]x^2+y^2=z^2[/tex], [tex]z>0[/tex]

    with the plane [tex]x+y+z=1[/tex], and let

    [tex]\textbf{f}(x,y,z)=<0,0,1/z^2>[/tex].

    I am trying to calculate [tex]\int_C \textbf{f} \wedge d\textbf{r}[/tex].

    2. Relevant equations

    Here [tex]\wedge[/tex] is the cross product in 3-space.

    3. The attempt at a solution

    I am able to solve the same sort of problem for the cylinder
    [tex]x^2+y^2=1[/tex]
    and the plane
    [tex]z=2y+1[/tex].
    In which case we let [tex]x = \cos t, y = \sin t[/tex] and [tex]z = 2\sin t +1[/tex] yeilding

    [tex]r(t) = <\cos t,\sin t,2\sin t +1>[/tex]

    from which the result follows from integration.

    I attempted to do something simular for the problem I am trying to figure out. Using [tex]r(t,z) = <z \cos t, z\sin t, z>[/tex] to represent points on the cone, I tried to move forward, but did not find a good way to combine this with the equation for the plane as I did in the above example for the cylinder. Perhaps it is because I have z in r(t,z)? Following this line of attack I can comebine this with the equation for the plane z = 1-x-y = 1-z cos t - sin t, solving for z and combining with r(t,z) yeilds r(t) as

    [tex]r(t) = (1+\cos t+\sin t)^{-1}<\cos t,\sin t,1>[/tex]

    but then I end up with horrible singularities!

    Would be most greatful if you would let me know how to solve this one!
     
    Last edited: Jun 8, 2009
  2. jcsd
  3. Jun 8, 2009 #2

    Dick

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    You meant to post x^2+y^2=z^2. x^2+y^2+z^2=z^2 is a line. The problem is that the hyperbola is unbounded and sin(t) and cos(t) are bounded functions. Try using their unbounded hyperbolic cousins, sinh(t) and cosh(t), in a similar way.
     
    Last edited: Jun 8, 2009
  4. Jun 8, 2009 #3
    Thanks, I fixed the equation you mention. I am not sure how I would use hyperbolic functions to represent the cone in the manor done with r(t,z) = <z cos t, z sin t, z>. I guess a hyperbola can be expressed as (+/- a cosh(u),b sinh(u)) but I am not sure how that helps me. Arg, this is confusing.
     
  5. Jun 8, 2009 #4
    Perhaps what I did is actually correct, privided I do not integrate past the singularities? This should not be so hard, but it is.
     
  6. Jun 8, 2009 #5

    Cyosis

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    Not entirely sure if this helps you but, the equation for a circle [itex]x^2+y^2=r^2[/itex] is parametrized by [itex]x=r \cos \theta, y =r \sin \theta[/itex], which you of course already know. Similarly the equation for a hyperbola x^2-y^2=r^2 is parametrized by [itex] x=r \cosh \theta, y=r \sinh \theta[/itex], note that [itex]\cosh^2 \theta-\sinh^2 \theta=1[/itex].
     
  7. Jun 8, 2009 #6

    Dick

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    Your posted quadratic is now a sphere. I still think you want x^2+y^2=z^2. If so try z=x*cosh(t) and y=x*sinh(t). That satisfies x^2+y^2=z^2 for any choice of x. Now substitute into the plane equation to find x in terms of t. It's really almost exactly the same thing you did with the cylinder example.
     
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