Total torsion and involutes of space curves

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Problem 3.1.14 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.1 "Differential Geometry of Curves"

(a)(b)

Call a normal vector field $v$ along a space curve $\gamma$ parallel if $v'$ is always tangent to $\gamma$ .

(a)

Show that the angle through which a parallel normal vector field $v$ turns relative to the principal normal $N$ along $\gamma$ is given by the total torsion of $\gamma$ , i.e.,

\theta (L) - \theta (0) = \int _0^L \tau (s) \, ds.

Here $s$ and $L$ denote arclength and the length of $\gamma$ , respectively.

(b)

Show that the total torsion of any closed curve $\gamma$ which lies on a sphere in $\mathbb {R}^{3}$ must vanish.

Problem 3.1.19 from Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. "Problems and Solutions in Mathematics" [CJP+11]More problems in Section 3.1 "Differential Geometry of Curves"

Let $\alpha$ and $\beta$ be two regular curves in $\mathbb {R}^{3}$ . The curve $\beta$ is called an involute of $\alpha$ if for all $t$ , $\beta (t)$ lies on the line tangent to $\alpha$ at $\alpha (t)$ and $\left\langle \alpha '(t), \beta '(t) \right\rangle = 0$ . Show that every involute of a generalized helix $\alpha$ is a plane curve. (Recall that $\alpha$ is a generalized helix if for some constant vector $u \neq 0$ , $\left\langle u, \alpha '(s) \right\rangle = \text{const}$ , where $s$ is arclength for $\alpha$ .)

References

[CJP+11] Chen J., Jiang G., Pan Y., Qin T., Tong Y., Wu Q., Xu S., Li T. Problems and Solutions in Mathematics. 2nd ed., World Scientific, 2011.