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Math Help - Parametrization of the intersection of cylinders

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    Parametrization of the intersection of cylinders

    Use sine and cosine to parametrize the intersection of the cylinders x^2+y^2=1 and x^2+z^2=1 (use two vector-valued functions). Then describe the projections of this curve on the three coordinate planes.

    I am not sure how to do this problem at all any help would be great. Thanks
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  2. #2
    Senior Member DeMath's Avatar
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    Re: Parametrization of the intersection of cylinders

    Quote Originally Posted by acasas4 View Post
    Use sine and cosine to parametrize the intersection of the cylinders x^2+y^2=1 and x^2+z^2=1 (use two vector-valued functions). Then describe the projections of this curve on the three coordinate planes.

    I am not sure how to do this problem at all any help would be great. Thanks
    The projection of the cylinder x^2+y^2=1 on the Oxy-plane is a circle x^2+y^2=1, which has this parametric equation \begin{cases}x=\cos{t},\\y=\sin{t},\end{cases}\!\!  t\in[0;2\pi].

    Also x^2+z^2=1~\Rightarrow~z=\pm\sqrt{1-x^2}=\pm\sqrt{1-\cos^2t}

    So, \begin{cases}x=\cos{t},\\y=\sin{t},\\z=\pm\sqrt{1-\cos^2t},\end{cases}\!\!t\in[0;2\pi].

    See the plot, where
    - the blue line is \left(\cos{t},\sin{t},-\sqrt{1-\cos^2t}\right),
    - the red line is \left(\cos{t},\sin{t},\sqrt{1-\cos^2t}\right).



    For Maple

    with(plots):
    A := spacecurve([cos(t), sin(t), sqrt(1-cos(t)^2)], t = 0..2*Pi, color=red, thickness=3):
    B := spacecurve([cos(t), sin(t), -sqrt(1-cos(t)^2)], t = 0..2*Pi, color=blue, thickness=3):
    display(A, B, axes=normal, scaling=constrained, view=[-2..2, -2..2, -2..2], numpoints=1000);
    Last edited by DeMath; October 13th 2011 at 10:05 AM.
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