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Geodesic curvature, normal curvature, and geodesic torsion

  1. Mar 3, 2014 #1
    I am struggling to make sense out some things. Hopefully someone can help or at least offer some different point of view. Let's examine a differential curve parameterized by arc length that maps some interval into an oriented surface (lets call it N(s)). The surface has a unit normal field restricted to the curve [itex]\alpha[/itex]. Also, let [itex]n(s):=N(s)\wedge T(s)[/itex] where [itex]T(s)=\alpha '(s)[/itex].

    If we define the derivatives of T, N, and n as the following

    [itex]T'=-k_{g}n+k_{n}N[/itex]
    [itex]n'=k_{g}T+\tau_{g}N[/itex]
    [itex]N'=-k_{n}T-\tau_{g}n[/itex]

    then we should have [itex]N'\cdot T = -k_{n}[/itex] and the second fundamental form is given by [itex]II(T,T) = k_{n}[/itex] while [itex]N'\cdot n=-\tau_{g}[/itex] so that the second fundamental form is given by [itex]II(T,n)=\tau_{g}[/itex].

    This seems pretty clear to me, unless I have my definitions mixed up some how. Does this seem correct?
     
  2. jcsd
  3. Mar 3, 2014 #2

    micromass

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    Hmm. I'm not sure what your ##N(s)## is supposed to be. Is ##N(s)## the normal of the surface at point ##\alpha(s)##?

    I get some sign differences. I think you should see http://en.wikipedia.org/wiki/Darboux_frame for the correct definitions.

    Not sure where this comes from. Could you clarify?
     
  4. Mar 3, 2014 #3
    Yes that is correct. I made a mistake in my initial statement. N(s) is the unit normal field to the surface at [itex]\alpha(s)[/itex]. Thanks for allowing me to clarify this. That means n(s) is the normal to the curve [itex]\alpha[/itex].

    I will look into this.

    The second fundamental form defined by [itex]II(v,v)=-\left\langle dN_{p}v,v\right\rangle[/itex] for any v in the tangent plane at a point p of S.
     
  5. Mar 3, 2014 #4

    micromass

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    I agree with that then, up to signs. (Not that signs are all that important)
     
  6. Mar 3, 2014 #5
    That is interesting, because I spoke with a professor recently about this and his claim was that the geodesic torsion was defined by [itex]II(T,N)=\tau_{g}[/itex]. But by inspection this wouldn't make sense. I'm just confused a little I guess.
     
  7. Mar 3, 2014 #6

    micromass

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    Signs are not so important anyway. If he defines his geodesic torsion like that, then I don't think it'll make a lot of difference. It'll yield the same theory up to sign.
     
  8. Mar 3, 2014 #7
    I agree with the sign issue but I think it does make a huge difference because [itex]N'\cdot n \ne N'\cdot N[/itex] by definition of the Darboux frame, and by that same definition [itex]N'\cdot n=\tau_{g}[/itex]. So [itex]II(T,N)\ne \tau_{g}[/itex].
     
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