- #1
Dahaka14
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Homework Statement
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].
Homework 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]
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.