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Differential Geometry Question

  1. Mar 30, 2009 #1
    1. The problem statement, all variables and given/known data
    Find an explicit unit-speed non-degenerate space curve [tex]\vec{r}:(0,\infinity)\rightarrow\Re^{3}[/tex] whose curvature and torsion [tex]\kappa,\tau:(0,\infinity)\rightarrow\Re[/tex] are given by the functions [tex]\kappa(s)=\tau(s)=\frac{1}{s}[/tex].

    2. Relevant equations
    the only thing that I can think of that would help us here are the Frenet equations:
    [tex]t'=\kappa n[/tex]
    [tex]n'=-\kappa t -\tau b[/tex]
    [tex]b'=\tau n[/tex]

    3. The attempt at a solution
    If we are to have [tex]\kappa(s)=\tau(s)=\frac{1}{s}[/tex], then we must have
    [tex]t'=\frac{1}{s} t[/tex] and
    [tex]b'=\frac{1}{s} t[/tex], thus
    [tex]t'=b'[/tex]. I'm not sure what to do after this point, as I messed with these equations for awhile to no avail.
     
  2. jcsd
  3. Mar 31, 2009 #2

    lanedance

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    Homework Helper

    hi dahaka14

    from your frenet equations you have
    [tex]\textbf{t}'=\kappa \textbf{n}[/tex]
    [tex]\textbf{b}'=-\tau \textbf{n}[/tex]

    write down a vector a, with some constants c & d we will choose
    [tex]\textbf{a}= c\textbf{t} + d \textbf{n}[/tex]

    differentiating
    [tex]\textbf{a}'= c.\textbf{t}' + d .\textbf{b}'= c .\kappa .\textbf{n} - d.\tau .\textbf{n} = \frac{1}{s} (c-d) \textbf{n}[/tex]

    so choose c=d and the vector a is constant, might as well make a a unit vector so set:
    [tex]c = d = \frac{1}{\sqrt{2}}[/tex]

    now think about the dot product of a with t and what this means...
    hopefully this helps you get started...
     
  4. Apr 2, 2009 #3
    The dot product should give
    [tex]\textbf{a}\cdot\textbf{t}=c\textbf{t}\cdot\textbf{t}=\frac{\textbf{t}\cdot\textbf{t}}{\sqrt{2}}[/tex]

    I'm not sure where to go from here. The only thing that I have been able to think of is that perhaps the curve should be a helix, since a helix is such that [tex]\frac{\tau}{\kappa}[/tex] is constant.

    Edit: that LaTeX image should have:
    torsion/curvature=constant
     
  5. Apr 2, 2009 #4

    lanedance

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    Homework Helper

    yeah i think you are on the right track, as i understand it a general helix is defined as when [tex]\frac{\tau}{\kappa}[/tex] is constant, which is equivalent to the tangent vector making a constant angle with some vector, say a, which is what your dot product shows as t.t = 1

    Not 100% where to go, but picking an aribtrary (a), then for s=0, a starting t which matches your dot product could be a good place to start
     
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