# 曲线的微分几何

（重定向自曲率向量

## 定义

${\displaystyle n}$  是一个正整数，${\displaystyle r}$  是正整数或 ${\displaystyle \infty }$ ${\displaystyle I}$  是实数非空区间，${\displaystyle t}$  属于 ${\displaystyle I}$ 。一个${\displaystyle C^{r}}$  类（即 ${\displaystyle \gamma }$ ${\displaystyle r}$ 连续可微向量值函数

${\displaystyle \mathbf {\gamma } :I\to {\mathbb {R} }^{n}}$

${\displaystyle \gamma :[a,b]\rightarrow \mathbb {R} ^{n}}$

${\displaystyle \lbrace \gamma '(t),\gamma ''(t),...,\gamma ^{(m)}(t)\rbrace {\mbox{, }}m\leq k}$

${\displaystyle \mathbb {R} ^{n}}$ 线性无关

${\displaystyle \gamma '(t)\neq 0}$  对任何 ${\displaystyle t\in I\,.}$

## 重新参数化与等价关系

${\displaystyle \mathbf {\gamma _{1}} :I_{1}\to R^{n}}$

${\displaystyle \mathbf {\gamma _{2}} :I_{2}\to R^{n}}$

${\displaystyle \phi :I_{1}\to I_{2}}$

${\displaystyle \phi '(t)\neq 0\qquad (t\in I_{1})}$

${\displaystyle \mathbf {\gamma _{2}} (\phi (t))=\mathbf {\gamma _{1}} (t)\qquad (t\in I_{1})\,.}$

γ2 称为 γ1重新参数化。这种 γ1 的重新参数化在所有参数 Cr 曲线的集合上定义了一种等价关系，其等价类称为 Cr 曲线

## 长度与自然参数化

C1 曲线 γ : [a, b] → Rn 的长度 l 可以定义为

${\displaystyle l=\int _{a}^{b}\vert \mathbf {\gamma } '(t)\vert dt.}$

${\displaystyle s(t)=\int _{t_{0}}^{t}\vert \mathbf {\gamma } '(x)\vert dx.}$

${\displaystyle {\overline {\mathbf {\gamma } (s)}}=\gamma (t(s))}$

${\displaystyle \vert {\overline {\mathbf {\gamma } '(s(t))}}\vert =1\qquad (t\in I).}$

${\displaystyle E(\gamma )={\frac {1}{2}}\int _{a}^{b}\vert \mathbf {\gamma } '(t)\vert ^{2}dt}$

## Frenet 标架

${\displaystyle \mathbf {e} _{1}(t),\ldots ,\mathbf {e} _{n}(t)}$

${\displaystyle \mathbf {e} _{1}(t)={\frac {\mathbf {\gamma } '(t)}{\|\mathbf {\gamma } '(t)\|}}}$
${\displaystyle \mathbf {e} _{j}(t)={\frac {{\overline {\mathbf {e} _{j}}}(t)}{\|{\overline {\mathbf {e} _{j}}}(t)\|}}{\mbox{, }}{\overline {\mathbf {e} _{j}}}(t)=\mathbf {\gamma } ^{(j)}(t)-\sum _{i=1}^{j-1}\langle \mathbf {\gamma } ^{(j)}(t),\mathbf {e} _{i}(t)\rangle \,\mathbf {e} _{i}(t)}$

${\displaystyle \chi _{i}(t)={\frac {\langle \mathbf {e} _{i}'(t),\mathbf {e} _{i+1}(t)\rangle }{\|\mathbf {\gamma } ^{'}(t)\|}}}$

Frenet 标架和广义曲率在重新参数化下是不变的，故它们是曲线的微分几何性质。

## 特殊 Frenet 向量和广义曲率

### 切向量

${\displaystyle \gamma '(t_{0})={\frac {d}{d\,t}}\mathbf {\gamma } (t),{t=t_{0}}}$

${\displaystyle \|\mathbf {\gamma } '(t_{0})\|}$

${\displaystyle \mathbf {e} _{1}(t)={\frac {\mathbf {\gamma } '(t)}{\|\mathbf {\gamma } '(t)\|}}.}$

${\displaystyle \mathbf {e} _{1}(s)=\mathbf {\gamma } '(s).}$

### 法向量

${\displaystyle {\overline {\mathbf {e} _{2}}}(t)=\mathbf {\gamma } ''(t)-\langle \mathbf {\gamma } ''(t),\mathbf {e} _{1}(t)\rangle \,\mathbf {e} _{1}(t).}$

${\displaystyle \mathbf {e} _{2}(t)={\frac {{\overline {\mathbf {e} _{2}}}(t)}{\|{\overline {\mathbf {e} _{2}}}(t)\|}}.}$

t 点的切向量和法向量张成 t 点的密切平面

### 曲率

${\displaystyle \kappa (t)=\chi _{1}(t)={\frac {\langle \mathbf {e} _{1}'(t),\mathbf {e} _{2}(t)\rangle }{\|\mathbf {\gamma } ^{'}(t)\|}}}$

${\displaystyle {\frac {1}{\kappa (t)}}}$

${\displaystyle \kappa (t)={\frac {1}{r}}\,,}$

### 次法向量

${\displaystyle \mathbf {e} _{3}(t)={\frac {{\overline {\mathbf {e} _{3}}}(t)}{\|{\overline {\mathbf {e} _{3}}}(t)\|}}\quad {\overline {\mathbf {e} _{3}}}(t)=\mathbf {\gamma } '''(t)-\langle \mathbf {\gamma } '''(t),\mathbf {e} _{1}(t)\rangle \,\mathbf {e} _{1}(t)-\langle \mathbf {\gamma } '''(t),\mathbf {e} _{2}(t)\rangle \,\mathbf {e} _{2}(t)}$

${\displaystyle \mathbf {e} _{3}(t)=\mathbf {e} _{2}(t)\times \mathbf {e} _{1}(t)\,.}$

### 挠率

${\displaystyle \tau (t)=\chi _{2}(t)={\frac {\langle \mathbf {e} _{2}'(t),\mathbf {e} _{3}(t)\rangle }{\|\mathbf {\gamma } '(t)\|}}}$

## 曲线论主要定理

${\displaystyle \chi _{i}\in C^{n-i}([a,b]){\mbox{, }}1\leq i\leq n}$

${\displaystyle \chi _{i}(t)>0{\mbox{, }}1\leq i\leq n-1}$

${\displaystyle \|\gamma '(t)\|=1{\mbox{ }}(t\in [a,b])}$
${\displaystyle \chi _{i}(t)={\frac {\langle \mathbf {e} _{i}'(t),\mathbf {e} _{i+1}(t)\rangle }{\|\mathbf {\gamma } '(t)\|}}\,,}$

${\displaystyle \mathbf {e} _{1}(t),\ldots ,\mathbf {e} _{n}(t)}$

${\displaystyle \mathbf {\gamma } (t_{0})=\mathbf {p} _{0}}$
${\displaystyle \mathbf {e} _{i}(t_{0})=\mathbf {e} _{i}{\mbox{, }}1\leq i\leq n-1}$

## Frenet-Serret 公式

Frenet-Serret 公式是一组一阶常微分方程。其解为由广义曲率函数 χi 所刻画的曲线的 Frenet 向量组。

### 2-维

${\displaystyle {\begin{bmatrix}\mathbf {e} _{1}'(t)\\\mathbf {e} _{2}'(t)\\\end{bmatrix}}={\begin{bmatrix}0&\kappa (t)\\-\kappa (t)&0\\\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{1}(t)\\\mathbf {e} _{2}(t)\\\end{bmatrix}}}$

### 3-维

${\displaystyle {\begin{bmatrix}\mathbf {e} _{1}'(t)\\\mathbf {e} _{2}'(t)\\\mathbf {e} _{3}'(t)\\\end{bmatrix}}={\begin{bmatrix}0&\kappa (t)&0\\-\kappa (t)&0&\tau (t)\\0&-\tau (t)&0\\\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{1}(t)\\\mathbf {e} _{2}(t)\\\mathbf {e} _{3}(t)\\\end{bmatrix}}}$

### n 维一般公式

${\displaystyle {\begin{bmatrix}\mathbf {e} _{1}'(t)\\\vdots \\\mathbf {e} _{n}'(t)\\\end{bmatrix}}={\begin{bmatrix}0&\chi _{1}(t)&&0\\-\chi _{1}(t)&\ddots &\ddots &\\&\ddots &0&\chi _{n-1}(t)\\0&&-\chi _{n-1}(t)&0\\\end{bmatrix}}{\begin{bmatrix}\mathbf {e} _{1}(t)\\\vdots \\\mathbf {e} _{n}(t)\\\end{bmatrix}}}$

## 参考文献

• Erwin Kreyszig, Differential Geometry, Dover Publications, New York, 1991, ISBN 9780484667218. Chapter II is is a classical treatment of Theory of Curves in 3-dimensions.
• 陈维桓，微分几何，北京大学出版社，北京，2006年，ISBN 7-301-10709.