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

Question

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

score 13 · Accepted Answer · answered Jul 30 '13 at 14:31

13

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.

answered Jul 30 '13 at 14:31

Ted Shifrin

115,160

But here I don't have $r$, how can I get $λ,μ,ν$ ? – JSCB 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... – JSCB Aug 01 '13 at 03:45
No, you're not using definitions of the Frenet frame. What is $r'$? – Ted Shifrin Aug 01 '13 at 05:10
Oh, let me see... – JSCB Aug 01 '13 at 06:06
$r’=(λ’-μκ)T+(μ’+λκ-ντ)N+(ν’+μτ)B$. If I then use $r\cdot r'=0$, I'll still get $λλ′+μμ′+νν′=0$. – JSCB Aug 01 '13 at 06:40
Yes, you'll need this next. But you're missing the most obvious: $r'=T$! – Ted Shifrin Aug 01 '13 at 06:49
I GET IT!! THANKS ~ – JSCB Aug 01 '13 at 07:01
2

Awesome! Very powerful technique, and analogues for surface theory too. If you feel like it, accept the answer so the question doesn't appear still to be open. – Ted Shifrin Aug 01 '13 at 12:47

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

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