I need some help with the following problem of differential geometry:

Suppose $\alpha$ is a unit speed curve with curvature $\kappa>0$ and torsion $\tau\neq 0$.

$\bf (a)$ If $\alpha$ lies on a sphere of center $c$ and radius $r$ show that $$\alpha-c=-\rho N-\rho^{'}\sigma B,$$ where $\rho=1/\kappa$ and $\sigma=1/\tau$. Conclude that $$r^2=\rho^2+(\rho^{'}\sigma)^2.$$

$\bf (b)$ Conversely, if $\rho^2+(\rho^{'} \sigma)^2$ has a constant value $r^2$ show that $\alpha$ lies on a sphere of radius $r$. (Hint: Use $(a)$ to define the center).

Any help will be welcome.

  • $\begingroup$ I have to prove this exact problem! Was just about to ask a question about it when I found yours. Have spent hours on it so far.. Any luck? $\endgroup$ – DeathByTensors Mar 12 '17 at 18:20
  • $\begingroup$ Sorry, it's been a long time since I've asked the question. I don't recall if I found the answer, probably I didn't find it otherwise I would have posted it. $\endgroup$ – PtF Mar 12 '17 at 22:10

Hint: For the first one, what you know is that the function $\lVert \alpha - c \rVert^2$ is constant. That is, its derivative vanishes. This tells you for instance that $$0 = T \cdot (\alpha - c).$$ Now, since $\kappa$ and $\tau$ are obtained through (sufficiently many) differentiations of $\alpha$, all you can really do here is to differentiate until you get what you want (I think you need to do it five times or so, so some patience is required); the Frenet–Serret formulas will help you simplify intermediate expressions.

For the second one, try $c = \alpha + \rho N + \rho' \sigma B$.


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