设$S$是三维欧氏空间中的一个参数曲面，$\boldsymbol p = P(u, v)$是$S$上的一个点，$\boldsymbol n$是$S$上的单位法向量，则：
\[\begin{aligned} \frac{\partial \boldsymbol n}{\partial u} &= \frac{FM - GL}{EG - F^2}\frac{\partial \boldsymbol p}{\partial u} + \frac{FL - EM}{EG - F^2}\frac{\partial \boldsymbol p}{\partial v} \\ \frac{\partial \boldsymbol n}{\partial v} &= \frac{FN - GM}{EG - F^2}\frac{\partial \boldsymbol p}{\partial u} + \frac{FM - EN}{EG - F^2}\frac{\partial \boldsymbol p}{\partial v} \end{aligned}\]其中$E, F, G$是$S$的first fundamental form的系数，$L, M, N$是其second fundamental form的系数：
\[\begin{aligned} E &= \left| \frac{\partial \boldsymbol p}{\partial u} \right|^2 \\ F &= \frac{\partial \boldsymbol p}{\partial u} \cdot \frac{\partial \boldsymbol p}{\partial v} \\ G &= \left| \frac{\partial \boldsymbol p}{\partial v} \right|^2 \\ L &= \boldsymbol n \cdot \frac{\partial^2\boldsymbol p}{\partial u^2} \\ M &= \boldsymbol n \cdot \frac{\partial^2\boldsymbol p}{\partial u\partial v} \\ N &= \boldsymbol n \cdot \frac{\partial^2\boldsymbol p}{\partial v^2} \end{aligned}\]PREVIOUS四元数基础