曲线的球面标线
曲线 $\Gamma:\ \gamma=\gamma(t)$ 的球面标线是指由曲线的切向量、主法线向量和副法线向量的末端点形成的这三种曲线, 由于它们均是单位向量, 故都位于单位球面上. 若 $s$ 是曲线 $\gamma$ 的弧长参数, 则它们是
切线球面标线 $\Gamma_1:\ \gamma_1=\vec{v}(s)=e_1(s)=\dot{\gamma}(s)$.
主法线球面标线 $\Gamma_2:\ \gamma_2=\vec{n}(s)=e_2(s)=\frac{\ddot{\gamma}(s)}{|\ddot{\gamma}(s)|}$.
副法线球面标线 $\Gamma_3:\ \gamma_3=\vec{b}(s)=e_3(s)=\frac{\dot{\gamma}(s)\times\ddot{\gamma}(s)}{|\ddot{\gamma}(s)|}$.
若 $s_1,s_2,s_3$ 分别记指 $\gamma_1$, $\gamma_2$, $\gamma_1$ 的弧长参数, 则有
\[ds_1=\kappa(s)ds,\quad ds_2=\sqrt{\kappa^2(s)+\tau^2(s)}ds,\quad ds_3=|\tau(s)|ds.\]
这三条曲线的曲率分别是
\[\kappa_1=\sqrt{1+(\frac{\tau}{\kappa})^2},\quad\kappa_2=\sqrt{1+\frac{(\kappa\dot{\tau}-\dot{\kappa}\tau)^2}{(\kappa^2+\tau^2)^3}},\quad\kappa_3=\sqrt{1+(\frac{\kappa}{\tau})^2}.\]
挠率分别是
\[\tau_1=\frac{\kappa\dot{\tau}-\dot{\kappa}\tau}{\kappa(\kappa^2+\tau^2)},\]
\[\tau_2=\frac{(\kappa^2+\tau^2)\frac{d}{ds}(\kappa\dot{\tau}-\dot{\kappa}\tau)-\frac{3}{2}(\kappa\dot{\tau}-\dot{\kappa}\tau)\frac{d}{ds}(\kappa^2+\tau^2)}{(\kappa^2+\tau^2)^3+(\kappa\dot{\tau}-\dot{\kappa}\tau)^2},\]
\[\tau_3=\frac{\kappa\dot{\tau}-\dot{\kappa}\tau}{\tau(\kappa^2+\tau^2)}.\]