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Writing the equations of a surface of revolution

  1. Feb 18, 2007 #1
    1. The problem statement, all variables and given/known data
    Write the equations of a surface of revolution with axis OZ:

    A) the Torus obtained by a rotation of a circle x= a + b*sin(u), y= 0, z = b*sin(u)

    0 < b < a

    B) the pseduosphere obtained by the rotation of a tractrix x= a*sin(u), y=0, z= a*(log(tan(u/2) + cos(u))

    C) the catenoid obtained by the rotation of the catenary x = a*cosh(u/a) y=0, z= u.

    2. Relevant equations
    Unfortuantly my professor gave the questions from another text, whose 1st chapter covers this material; however, it is not covered in my material (he claims that we should have enough intuition to figure this out, however, I don't seem to have it.

    But besides that point, I was able to find the following general standard form of parametrization of a surface of revolution on mathworld's website:

    x(u,v)= phi(v)*cos(u)
    y(u,v)= phi(v)*sin(u)
    z(u,v)= psi(v)

    3. The attempt at a solution

    As hinted at above, I don't quite have the background to tackle this problem. At the moment I have written down the standard form, as above; however, I don't know what the functions phi or psi are, nor do I have much of an idea of how to find them, so if there are any ideas of where to go with this.

    Or I could be looking at this the entirely wrong way. So any help would be appreached so I can get this started.

    (Oh and the website that I snaged the equation from is as follows:

    http://mathworld.wolfram.com/SurfaceofRevolution.html)

    Thanks in advanced for any suggestions on where to get started.
     
  2. jcsd
  3. Feb 19, 2007 #2

    HallsofIvy

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    Staff Emeritus
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    Remember that a surface can be given as parametric equations with two parameters: x(u,v), y(u,v), z(u,v).

    In the first problem, "A) the Torus obtained by a rotation of a circle x= a + b*sin(u), y= 0, z = b*sin(u) " you are already given a parameter u. Since you are rotating around the z-axis, let v be the angle made with the x-axis. Obviously z will not change but x= r cos(v), y= r sin(v) where r is the distance in the xz-plane (in other words, x).
     
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