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I want to prove that unit speed spherical curve $\beta$ satisfies following inequality $$\kappa_{\beta}(s)\geqslant \frac1{R},$$ where $\kappa$ is curvature and $R$ is the radius of the sphere, that $\beta$ lies on.

How should I approach this problem?

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closed as off-topic by colormegone, zz20s, Ben Sheller, user296602, JKnecht Apr 26 '16 at 0:03

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Let $\vec{x}(s), \hat{t}(s), \hat{n}(s)$ be the position, tangent and normal vectors at arc length parametrization $s$.

Start from the equation of sphere $|\vec{x}(s)|^2 = R^2$, differentiate it once give us $$\hat{t}(s)\cdot\vec{x}(s) = 0$$ Differentiate it again, we have $$\hat{t}(s)\cdot\hat{t}(s) + \frac{d\hat{t}(s)}{ds}\cdot\vec{x}(s) = 0 \quad\implies\quad \kappa(s) \hat{n}(s)\cdot\vec{x}(s) = - 1$$

This leads to $$R = |\vec{x}(s)| \ge |\hat{n}(s)\cdot\vec{x}(s)| = \frac{1}{\kappa(s)} \iff \kappa(s) \ge \frac{1}{R}$$

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  • $\begingroup$ Why does $|\vec{x}(s)|\geqslant |\hat{h}(s)\cdot \vec{x}(s)|$ hold? $\endgroup$ – user334504 Apr 25 '16 at 15:02
  • $\begingroup$ @user334504 $\hat{n}(s)$ is a unit vector, so $\hat{n}(s)\cdot\vec{x}(s) = |\hat{n}(s)||\vec{x}(s)|\cos\theta = |\vec{x}(s)|\cos\theta$ where $\theta$ is the angle between $\hat{n}$ and $\vec{x}$. $\endgroup$ – achille hui Apr 25 '16 at 15:12

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