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can someone PLEASE let me know what I am doing wrong here? I feel like I'm missing something very basic and it's driving me crazy.

The question asks me to show that if both curvature and torsion of a unit speed spherical curve with center c and radius R are nowhere vanishing, then

(1) $\dfrac{1}{\kappa^2} + (\dfrac{1}{\tau} \dfrac{d}{ds} (\dfrac{1}{\kappa}))^2 = R^2$

and

(2) $\dfrac{\tau}{\kappa} + \dfrac{d}{ds} ( \dfrac{1}{\tau} \dfrac{d}{ds} (\dfrac{1}{\kappa})) = 0$

I have from a previous problem that for non-vanishing curvature,

$(q-c) N = -\dfrac{1}{\kappa}$

and

$\tau (q-c) B = \dfrac{d}{ds} (\dfrac{1}{\kappa})$

(where $\tau$ is torsion, $B$ is unit binormal, $N$ is unit normal, and $\kappa$ is curvature.

So here's what I did to try to obtain (1):

$\dfrac{1}{\kappa^2} + (\dfrac{1}{\tau} \dfrac{d}{ds} (\dfrac{1}{\kappa}))^2 = ((q-c)N)^2 + ((q-c) B)^2 = 2R^2$

I similarly cannot obtain (2).

Please help!!!! Thank you.

Mary Ogawa
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    It is already answered in http://math.stackexchange.com/questions/509598/arc-length-parameterization-lying-on-a-sphere/509747#509747 – Michael Hoppe Mar 13 '16 at 18:28
  • Thanks, I see it. But can you tell me what is wrong with my attempt? – Mary Ogawa Mar 13 '16 at 19:11
  • What you've written (regarding the results of a previous problem) basically does not make sense — you're setting a vector equal to a scalar, for starters. – Ted Shifrin Mar 14 '16 at 01:12

0 Answers0