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Math Help - Intersection of surfaces

  1. #1
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    Intersection of surfaces

    I want to find the equation of the intersection between a sphere and cylinder (in the first octant) but it's kind of weird.

    Sphere: x^2 + y^2 +z^2 = 4
    Cylinder: x^2 + y^2 - 2y = 0

    If I just sub one of them into the other I get:
    2y + z^2 = 4

    but that doesn't make sense to me since that is an equation of a surface since x can vary. There should be some sort of restriction on x... but how do I get this?
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  2. #2
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    Quote Originally Posted by eddi View Post
    I want to find the equation of the intersection between a sphere and cylinder (in the first octant) but it's kind of weird.

    Sphere: x^2 + y^2 +z^2 = 4
    Cylinder: x^2 + y^2 - 2y = 0

    If I just sub one of them into the other I get:
    2y + z^2 = 4

    but that doesn't make sense to me since that is an equation of a surface since x can vary. There should be some sort of restriction on x... but how do I get this?
    The intersection will be a curve, not a surface, and that is best described parametrically. Write the equation of the cylinder as x^2 + (y-1)^2=1, and you see that this can be parametrised as x = \sin\theta,\:y = 1+\cos\theta. Then x^2 + y^2 = 2+2\cos\theta, and if you substitute that into the equation of the sphere you see that z^2 = 2(1-\cos\theta) = 4\sin^2(\theta/2).

    Therefore the part of the curve in the positive octant can be described by the parametric representation (x,y,z) = (\sin\theta, 1+\cos\theta, \sin(\theta/2))\;(0\leqslant\theta\leqslant\pi).
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  3. #3
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    Quote Originally Posted by Opalg View Post
    The intersection will be a curve, not a surface, and that is best described parametrically. Write the equation of the cylinder as x^2 + (y-1)^2=1, and you see that this can be parametrised as x = \sin\theta,\:y = 1+\cos\theta. Then x^2 + y^2 = 2+2\cos\theta, and if you substitute that into the equation of the sphere you see that z^2 = 2(1-\cos\theta) = 4\sin^2(\theta/2).

    Therefore the part of the curve in the positive octant can be described by the parametric representation (x,y,z) = (\sin\theta, 1+\cos\theta, \sin(\theta/2))\;(0\leqslant\theta\leqslant\pi).
    Thanks. So to reiterate (for myself), I have to parametrize the cylinder aka:
    x = sin(t)
    y = 1 + cos(t)
    z = p

    Then transform my sphere to use t and p:
    x^2 + y^2 + z^2 = 4 turns into sin(t)^2 + (1+cos(t))^2 + p^2 = 4.

    Then I can solve for p in terms of t, substitute that back into the parametrization for the cylinder, and voila I have a set of parametric equations describing my curve. Haha cool.
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