# Show that $r$ is a spherical curve iff $(1/\kappa)^2+((1/\kappa)'(1/\tau))^2$ is a constant.

Let $r(s)$ be a curve parametrized by arc length, and $\kappa,\kappa',\tau$ are non-zero. Show that $r$ is a spherical curve iff $(1/\kappa)^2+((1/\kappa)'(1/\tau))^2$ is a constant.

The teacher gave the hint "center = $r+(1/\kappa)N+(1/\kappa)'(1/\tau)B$". I know how to get the proof from this, but how can we show that this is the center? Please help. Thanks.

• Show that the distance $\| r(s) - c \|$, where $c$ is the center, is a constant. (Show that its derivative is zero.) – Myself Jul 30 '13 at 14:37

Big Hint: The standard technique in all such differential geometry problems is to write $$r = \lambda T + \mu N + \nu B$$ for some functions $\lambda$, $\mu$, and $\nu$. Differentiate, use what you're given, and use the Frenet equations and use the fact that $T,N,B$ form a basis to get three equations.
• But here I don't have $r$, how can I get $λ,μ,ν$ ? – A. Chu Jul 30 '13 at 14:47
• You don't get them explicitly, but you get equations relating them and their derivatives if you follow my suggestion. The crucial thing is: What do you get if you assume the center of the sphere is $0$ and you differentiate $r\cdot r$? – Ted Shifrin Jul 30 '13 at 14:53
• I get $λλ'+μμ'+νν'=0$, but I really can't see how I should proceed to the result... – A. Chu Aug 1 '13 at 3:45
• No, you're not using definitions of the Frenet frame. What is $r'$? – Ted Shifrin Aug 1 '13 at 5:10