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Math Help - patch of a torus

  1. #1
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    Question patch of a torus

    Let C be the circle in the x-z plane with radius r>0 and center (R,0,0). A torus of revolution is obtained by revolving C about the z-axis. Show that the patch is given by x(u,v)=((R+r cos u)cos v, (R+r cos u)sin v, r sin u).

    Ahmmm, a patch is just a mapping x: R^{2} to R^{3} that defines the whole curve.
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  2. #2
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    Quote Originally Posted by wiz_girl View Post
    Let C be the circle in the x-z plane with radius r>0 and center (R,0,0). A torus of revolution is obtained by revolving C about the z-axis. Show that the patch is given by x(u,v)=((R+r cos u)cos v, (R+r cos u)sin v, r sin u).

    Ahmmm, a patch is just a mapping x: R^{2} to R^{3} that defines the whole curve.
    I've attached a drawing how to calculate the coordinates of the points of the surface of a torus.

    1. The black circle in the x-y-plane is the path of point R. In the picture of the torus I've choosen R=5

    2. The red circle has the radius \rho = R+r \cdot \cos(u). I've taken r = 3.

    With \rho and the angle v you can calculate the x and y-coordinate of the points.

    3. The z-coordinate of all points only depends on r and the angle u.
    Attached Thumbnails Attached Thumbnails patch of a torus-torus3.gif  
    Last edited by earboth; January 29th 2008 at 05:20 AM.
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