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Intersection of cylinder and plane

  1. Jan 14, 2006 #1
    I'm trying to find the parameterization of the intersection of a cylinder x^2+y^2=1 and the plane x+y+z=1, but I'm not exactly sure how to go about it. Any guidance on how to find this intersection in a parameterized form would be most appreciated.

    In general I don't know a great deal about finding intersections of various surfaces and shapes in r^3 or how to parameterize these things. Googling hasn't turned up anything particularly usefull, but a few scattered examples. I was also wondering if anyone knew of useful sites with this information.

  2. jcsd
  3. Jan 14, 2006 #2


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    parameterization of cylinder and plane intersection

    The parameterization of the cylinder [itex]x^2+y^2=1[/itex] is standard:

    Let x(t)=cos(t) and let y(t)=cos(t) for [itex]0\leq t < 2\pi[/itex].

    We wish to parameterize the intersection of the above cylinder and the plane x+y+z=1, solving this for z gives z=1-x-y so we see that if we put

    z(t) = 1-x(t)-y(t) = 1-cos(t)-sint(t) for [itex]0\leq t < 2\pi[/itex],

    then the parameterization we seek is given by:

    [tex] \vec{r}(t) = \left< x(t),y(t),z(t)\right> = \left< \cos (t),\sin (t), 1- \cos (t) - \sin (t)\right> ,\mbox{ for }0\leq t < 2\pi[/tex].

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