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.