问题及解答

写出圆柱螺旋曲线(circular helix) $\gamma(t)=(a\cos t,a\sin t,ct)$, $a > 0$, $c\neq 0$ 的 Serret-Frenet 标架 $\vec{v},\vec{n},\vec{b}$. 并求其曲率和挠率.

回忆公式(见问题724):

\[
\begin{aligned}
\vec{v}&=\frac{\dot{\gamma}(t)}{|\dot{\gamma}(t)|},\\
\vec{n}&=\frac{(\dot{\gamma}(t)\times\ddot{\gamma}(t))\times\dot{\gamma}(t)}{|(\dot{\gamma}(t)\times\ddot{\gamma}(t))\times\dot{\gamma}(t)|},\\
\vec{b}&=\frac{\dot{\gamma}(t)\times\ddot{\gamma}(t)}{|\dot{\gamma}(t)\times\ddot{\gamma}(t)|}.
\end{aligned}
\]

计算

\[
\begin{aligned}
\dot{\gamma}(t)&=(-a\sin t,a\cos t,c),\\
\ddot{\gamma}(t)&=(-a\cos t,-a\sin t,0),\\
\end{aligned}
\]

\[
\begin{split}
\dot{\gamma}(t)\times\ddot{\gamma}(t)&=
\begin{vmatrix}
\vec{i} & \vec{j} & \vec{k}\\
-a\sin t & a\cos t & c\\
-a\cos t & -a\sin t & 0
\end{vmatrix}\\
&=(ac\sin t)\vec{i}-(ac\cos t)\vec{j}+a^2\vec{k}\\
&=a(c\sin t,-c\cos t,a),
\end{split}
\]

\[
\begin{split}
(\dot{\gamma}(t)\times\ddot{\gamma}(t))\times\dot{\gamma}(t)&=
\begin{vmatrix}
\vec{i} & \vec{j} & \vec{k}\\
ac\sin t & -ac\cos t & a^2\\
-a\sin t & a\cos t & c
\end{vmatrix}\\
&=((-ac^2-a^3)\cos t)\vec{i}-((ac^2+a^3)\sin t)\vec{j}+0\vec{k}\\
&=-a(a^2+c^2)(\cos t,\sin t,0).
\end{split}
\]

\[
\begin{aligned}
|\dot{\gamma}(t)|&=\sqrt{a^2+c^2},\\
|\dot{\gamma}(t)\times\ddot{\gamma}(t)|&=a\sqrt{a^2+c^2},\\
|(\dot{\gamma}(t)\times\ddot{\gamma}(t))\times\dot{\gamma}(t)|&=a(a^2+c^2),\\
\end{aligned}
\]

因此

\[
\begin{aligned}
\vec{v}(t)&=\frac{1}{\sqrt{a^2+c^2}}(-a\sin t,a\cos t,c),\\
\vec{n}(t)&=(-\cos t,-\sin t,0),\\
\vec{b}(t)&=\frac{1}{\sqrt{a^2+c^2}}(c\sin t,-c\cos t,a).\\
\end{aligned}
\]

回忆曲率和挠率的公式 (见问题724)

\[\kappa(t)=\frac{|\dot{\gamma}(t)\times\ddot{\gamma}(t)|}{|\dot{\gamma}(t)|^3},\qquad\tau(t)=\frac{\det(\dot{\gamma}(t),\ddot{\gamma}(t),\dddot{\gamma}(t))}{|\dot{\gamma}(t)\times\ddot{\gamma}(t)|^2}.\]

还需计算 $\dddot{\gamma}(t)$ 等.

\[\dddot{\gamma}(t)=(a\sin t,-a\cos t,0),\]

\[
\begin{split}
\det(\dot{\gamma}(t),\ddot{\gamma}(t),\dddot{\gamma}(t))&=\langle\dot{\gamma}(t)\times\ddot{\gamma}(t),\dddot{\gamma}(t)\rangle\\
&=\langle a(c\sin t,-c\cos t,a),(a\sin t,-a\cos t,0)\rangle\\
&=a^2 c.
\end{split}
\]

因此,

\[
\begin{split}
\kappa(t)&=\frac{|\dot{\gamma}(t)\times\ddot{\gamma}(t)|}{|\dot{\gamma}(t)|^3}\\
&=\frac{a\sqrt{a^2+c^2}}{(\sqrt{a^2+c^2})^3}\\
&=\frac{a}{a^2+c^2},
\end{split}
\]

\[
\begin{split}
\tau(t)&=\frac{\det(\dot{\gamma}(t),\ddot{\gamma}(t),\dddot{\gamma}(t))}{|\dot{\gamma}(t)\times\ddot{\gamma}(t)|^2}\\
&=\frac{a^2 c}{(a\sqrt{a^2+c^2})^2}\\
&=\frac{c}{a^2+c^2}.
\end{split}
\]