Prove that the trace of $\alpha$ is contained in a sphere of radius $r>0$ if and only if $\frac{1}{k(s)^2}+\frac{k'(s)^2}{k(s)^4\tau(s)^2}=r^2$

Asked Dec 06 '20 at 17:48

Active Dec 06 '20 at 17:48

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$\textbf{preliminaries}$

I am in the course of geometry of curves and surfaces, clearly frenet's trihedron is $\{T(s),N(s),B(s)\}$ where $T(s)=\alpha'(s)$ , $N(s)=\frac{T'(s)}{\|T'(s)\|}$ and $B(s)=T(s) \times N(s)$ The Serret-Frenet equations of the curve $\alpha$ are:

$$T'(s)=k(s)N(s)$$ $$N'(s)=-k(s)T(s)-\tau(s)B(s)$$ $$B'(s)=\tau(s)N(s)$$

$\textbf{Question}$ Let $\alpha: I \to \mathbb{R}^3$ be a curve p.b.a.l. With positive curvature such that $k'(s)\not = 0$ and $\tau(s)\not = 0$ for each $s\in I$. Prove that the trace of $\alpha$ is contained in a sphere of radius $r>0$ if and only if $$\frac{1}{k(s)^2}+\frac{k'(s)^2}{k(s)^4\tau(s)^2}=r^2$$

$\textbf{My attempt:}$

If the trace of $\alpha$ lies in a sphere of radius $r>0$, then $\|\alpha\|^2=r^2$ Differentiating this expression three times, we have: $$\langle \alpha,T \rangle=0$$ $$1+k\langle \alpha, N \rangle=0$$ and $$k'\langle \alpha, N\rangle -k\tau \langle \alpha, B \rangle=0$$ therefore: $r^2=\|\alpha\|^2=\langle \alpha , T \rangle^2+\langle \alpha, N \rangle^2+ \langle \alpha, B\rangle^2=(\frac{1}{k})^2+(\frac{k'}{\tau k^2})^2$.

Here I assumed the following equality $\|\alpha\|^2=\langle \alpha , T \rangle^2+\langle \alpha, N \rangle^2+ \langle \alpha, B\rangle^2$ but in reality I don't know how to justify it, can you tell me how to justify this equality?

conversely this turns out

$\textbf{imagge attached}$

asked Dec 06 '20 at 17:48

wessi

I'm confused - What is $\alpha$ and what is its trace? – K.defaoite Dec 06 '20 at 18:03
@K.defaoite $\alpha$ is a curve p.b.l.a, see my question. – wessi Dec 06 '20 at 18:04
What does p.b.l.a stand for? – K.defaoite Dec 06 '20 at 18:06
parameterized by arc length – wessi Dec 06 '20 at 18:06
This identical question was posted a few days ago. There are dozens of posts on spherical curves. Why don't you search the site before posting? – Ted Shifrin Dec 06 '20 at 18:15
@TedShifrin Give me the link of the identical question – wessi Dec 06 '20 at 18:16
There is one. There are lots, lots more. Just search. – Ted Shifrin Dec 06 '20 at 18:17
NO NO!! My question is just a detail you have ignored – wessi Dec 06 '20 at 18:18
@TedShifrin you should consider reading my question before closing other questions – wessi Dec 06 '20 at 18:19
1

And here is another one from a few days ago. https://math.stackexchange.com/questions/3933223/question-about-curvature/3933378#3933378 But you're right. Your question has nothing to do with differential geometry. Go review what it means to have an orthonormal basis for $\Bbb R^3$. – Ted Shifrin Dec 06 '20 at 18:24
and you keep making your videos on youtube – wessi Dec 06 '20 at 18:26

Prove that the trace of $\alpha$ is contained in a sphere of radius $r>0$ if and only if $\frac{1}{k(s)^2}+\frac{k'(s)^2}{k(s)^4\tau(s)^2}=r^2$

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