1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Surface of constant curvature

  1. Dec 15, 2008 #1
    1. The problem statement, all variables and given/known data

    I'm trying to find a surface of revolution with Gauss curvature K of +1 at all points, which doesn't lie in a sphere.

    2. Relevant equations

    The surface is parametrized as [itex]\psi (t, \theta ) = ( x(t), y(t) cos \theta , y(t) sin \theta ) [/itex]

    I have the equation
    K = \frac{x' (x'' y' - x' y'')}{y(x'^2 + y'^2)^2}

    3. The attempt at a solution

    I am thinking it has to do with the curve [itex] \alpha (t) = (x(t),y(t)) [/itex] not having unit speed, but I am kind of stuck as to where to go from there.

  2. jcsd
  3. Dec 15, 2008 #2


    User Avatar
    Science Advisor
    Homework Helper

    Is there one? The only way out I can think of is to make it disconnected. A surface of constant guassian curvature is locally isometric to a sphere.
  4. Dec 16, 2008 #3
    I know that for a unit-speed [itex]\alpha (t)[/itex], the equation reduces to [itex]K = \frac{-y''}{y}[/itex], which does clearly represent a sphere.

    The way the question is worded on my homework seems to point to the fact that that reduction only applies to unit-speed curves, which is why I think that perhaps an [itex]\alpha (t)[/itex] that doesn't have unit-speed perhaps can give a surface of revolution with constant curvature +1 that isn't a sphere... but maybe there's another "gimmick" that I'm overlooking...

    Anyways, thanks for your help!
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Surface of constant curvature
  1. Curvature Proofs. (Replies: 1)

  2. Curvature Proof (Replies: 1)

  3. Finding Curvature (Replies: 4)

  4. Compute the curvature (Replies: 15)

  5. Find the curvature? (Replies: 5)