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!

Semigeodesic parameterization I am a little confused

  1. Mar 10, 2007 #1
    Today I was working through a handout my professor gave me on geodesics and stumbled upon the section on semigeodesic parameterization...and then I got lost.

    I was able to follow the material through the point where the text points out that the first fundemental form will consist of:

    I= E(du)^2 + G*(dv)^2, since the parametrization is orthogonal. The text then moves into an example of how if v were constant the differential equation for the geodesic

    u''v'-u'v''+Av'-Bu' = 0 (where A and B's values can be derived from here: http://mathworld.wolfram.com/GeodesicCurvature.html )

    becomes

    -1/2*[tex]E_v[/tex]/G = 0.

    Which makes sense, but then it gives an alternative statement later on that the above equation of the geodesic could also be expressed as

    da/dv = - partial((G)^(1/2))/partial(u). Where a is defined as the angle the geodesics intersect the curves v=constant.

    Whats worse is this alternative to the above was offered in the "theorems" section of the chapter and doesn't have a proof with it.

    So two questions:

    1) Wouldn't the angle be pi/2, since in a semigeodesic parameterization any geodesic would be orthogonal to the to the coordinate curves (in this case v=constant)?

    2) Where are they getting the alternative expression...is is from Gauss-Bonnet? Or am I just reading the theorem incorrectly. In any case, where the heck is the text coming up with said "theorem." I don't want a proof, I just want a reasonable feel for where the heck they are getting it from. Because from their definations I just don't see the correlation.
     
  2. jcsd
  3. Aug 18, 2009 #2

    Reb

    User Avatar



    1) Not in the case of a random manifold.

    2) Not really. It is derived by a change of variables in the pde [tex]E_v/2G=0[/tex].
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Semigeodesic parameterization I am a little confused
  1. Am I right? (Replies: 5)

Loading...