Results 1 to 6 of 6
Like Tree1Thanks
  • 1 Post By chiro

Math Help - Asymptotic curves

  1. #1
    Newbie
    Joined
    Nov 2012
    From
    utah
    Posts
    10

    Asymptotic curves

    Consider the helicoid S given by the parametrization x(u,v)=(vcosu, vsinu,u). Find the asymptotic curves on S.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    4,163
    Thanks
    761

    Re: Asymptotic curves

    Hey julietteeden.

    What does the terminology asymptotic curve mean?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Nov 2012
    From
    utah
    Posts
    10

    Re: Asymptotic curves

    A curve is asymptotic if its normal curvature is 0. I found the normal curvature but I'm not sure how to find when it equals 0.
    normal curvature N=(-sinu,cosu,-v)/sqrt(1+v^2)
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    4,163
    Thanks
    761

    Re: Asymptotic curves

    When does N = 0?
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Nov 2012
    From
    utah
    Posts
    10

    Re: Asymptotic curves

    When u and v are constant?
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor
    Joined
    Sep 2012
    From
    Australia
    Posts
    4,163
    Thanks
    761

    Re: Asymptotic curves

    The curvature should be a numeric quantity (scalar) that is positive for a positively oriented space and negative for a negatively oriented space.

    Now if you are trying to find the curvature at a typical point where that equals zero then you will need the Bi-Normal, Normal, and Tangent vectors do this calculation.

    So you have the normalized length of the normal when you look at what the normalized vector is divided by (SQRT(1 + v^2))

    The only way I can see this happening is basically in line with your answer that they are both constant.

    If you have something with a zero curvature then it is a plane.

    In this case it is a plane parametrized by two parameters u and v and a plane is flat if the tangent vectors for the rate of change of how the plane changes in that direction is zero.

    Both tangential vectors (the primary tangent and the secondary tangent or bi-normal) will be constant for a flat space and this is only satisfied if your above conditional holds (i.e. u and v are constant).

    If they are allowed to vary then SQRT(1 + v^2) >= 1 for all v so it will never approach zero for all real values of v. If you are considering complex values then this is even more complex (this is a branch known as Kahler geometry with an umlaut a) with completely different conditions.

    Now this is for the curvature at a point: if you need to consider other forms of curvature then you will need to consider what those are, but for flat objects the intrinsic curvature as far as I recall should be 0 as well.
    Thanks from julietteeden
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Asymptotic theory
    Posted in the Advanced Statistics Forum
    Replies: 0
    Last Post: March 28th 2012, 03:59 PM
  2. Asymptotic
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: March 30th 2011, 01:02 AM
  3. [SOLVED] n*log(n) asymptotic behaviour
    Posted in the Algebra Forum
    Replies: 0
    Last Post: June 29th 2010, 07:58 AM
  4. asymptotic bounds
    Posted in the Discrete Math Forum
    Replies: 0
    Last Post: May 8th 2010, 06:20 PM
  5. sketching curves-transformation curves f(x)!!!!
    Posted in the Pre-Calculus Forum
    Replies: 3
    Last Post: December 2nd 2006, 07:55 AM

Search Tags


/mathhelpforum @mathhelpforum