# (I) 參數曲線
## 參數曲線
==$這裡只考慮\mathbb{R}^3裡的曲線!!$==
$曲線可以被定義成一個由區間I=(a,b)到\mathbb{R}^3的映射:$
:::success
$\alpha:I\subseteq\mathbb{R}\to\mathbb{R}^3,其中I=(a,b)$
:::
$這個映射的像稱為$==$曲線的軌跡$==$:$
:::success
$I_\alpha\overset{def}{=}\{(x(t),y(t),z(t))|t\in I\}$
:::
## 可微參數曲線
$可微的定義:$
:::success
$\alpha(t)可微(C^\infty),即 x(t),y(t),z(t)皆可微(C^\infty)$
:::
$此時\alpha^\prime(t)(也就是\dfrac{d\alpha}{dt})稱為$==$\alpha的切向量$==
### [e.g.]1.圓

$\begin{aligned}&\alpha:(-\varepsilon,2\pi+\varepsilon)\to\mathbb{R}^2,\alpha(t)=(a\cos(t),a\sin(t)),a>0
\\&\because\begin{cases}x(t)=a\cos(t)\\y(t)=a\sin(t)\end{cases}\in C^\infty\therefore\alpha可微\end{aligned}$
::: danger
* $兩個軌跡相同但有不同「速率」、「方向」的曲線不相等$
$如(a\cos(t),a\sin(t))和(a\cos(2t),a\sin(2t))的軌跡都是x^2+y^2=a^2,$但兩者的切向量並不同，因此不相等
:::
### [e.g.]2.帶尖點的曲線
$\begin{aligned}
&\alpha:\mathbb{R}\to\mathbb{R}^2,\alpha(t)=(t^3,t^2)
\\&\because\begin{cases}x(t)=t^3\\y(t)=t^2\end{cases}\in C^\infty\therefore\alpha可微,又\begin{cases}x^\prime(t)=3t^2\\y^\prime(t)=2t\end{cases}在t=0時皆為0
\\&\implies\alpha在(0,0)有尖點(cusp)
\end{aligned}$
### [e.g.]3.絕對值函數
$\begin{aligned}
&\alpha:\mathbb{R}\to\mathbb{R}^2,\alpha(t)=(t,|t|)
\\&\because y(t)=|t|在t=0不可微\therefore\alpha不可微
\end{aligned}$
### [e.g.]4.帶自交點之曲線
$\begin{aligned}
&\alpha:\mathbb{R}\to\mathbb{R}^2,\alpha(t)=(t^3-t,t^2-1)
\\&\because\begin{cases}x(t)=t^3-t\\y(t)=t^2-1\end{cases}\in C^\infty\therefore\alpha可微
\end{aligned}$
::: danger
* $\alpha(t)不一定是一對一$
:::
## 正則
$正則的定義:$
:::success
$若\alpha^\prime(t)\not=0\ \forall t\in I,則稱\alpha是正則曲線$
:::
### [e.g.]螺線
$\begin{aligned}
&\alpha:\mathbb{R}\to\mathbb{R}^3,\alpha(t)=(a\cos(t),a\sin(t),bt),a,b\not=0
\\&\because\alpha^\prime(t)=(-a\sin(t),a\cos(t),b)\not=0\ \forall x\in\mathbb{R}
\\&\therefore\alpha(t)是正則曲線
\end{aligned}$
# (II) 弧長參數曲線
## 弧長的計算
$給定向量\vec{v}\in\mathbb{R}^3,其長度之計算方式如下:$
:::warning
$|\vec{v}|=\sqrt{v_x^2+v_y^2+v_z^2}$
:::
$現在在考慮一正則曲線\alpha:I\to\mathbb{R}^3和微小量\Delta t,\\則在\alpha(t)及\alpha(t+\Delta t)間的曲線可近似為一條直線:$

:::warning
$\Delta s\approx|\alpha(t+\Delta t)-\alpha(t)|=\Delta t\cdot|\dfrac{\alpha(t+\Delta t)-\alpha(t)}{\Delta t}|\approx|\alpha^\prime(t)|\Delta t$
:::
$將區間內所有小段累加並取極限,得到弧長:$
:::warning
$s=\displaystyle\int_I|\alpha^\prime(t)|dt=\int_a^b|\alpha^\prime(t)|dt$
:::
$得到對點t_0\in I的弧長公式:$
:::success
$s(t)=\displaystyle\int_{t_0}^t|\alpha^\prime(u)|du=\int_{t_0}^t\sqrt{x^\prime(u)^2+y^\prime(u)^2+z^\prime(u)^2}du$
:::
### [e.g.]圓


$\begin{aligned}
&\alpha:(-\varepsilon,2\pi+\varepsilon)\to\mathbb{R}^2,\alpha(t)=(a\cos(t),a\sin(t)),a>0
\\&s(t)=\displaystyle\int^t_0\sqrt{(-a\sin(u))^2+(a\cos(u))^2}du=\int^t_0a \ du=at,s(2\pi)=2\pi a
\end{aligned}$
## 以弧長參數化曲線(parametrize by arc-length)
$弧長參數曲線的定義:$
:::success
$\alpha(s)為正則且|\alpha^\prime(s)|=1者稱為弧長參數曲線$
:::
$也可以將一般的正則曲線調整成弧長參數曲線,思路如下:$
:::warning
$\begin{aligned}
&\alpha(s)=\alpha(t(s)),t(s)是t對弧長的函數
\\&|\alpha^\prime(s)|=|\dfrac{d\alpha}{dt}\cdot\dfrac{dt}{ds}|=1,即|\alpha^\prime(t)|=|\dfrac{ds}{dt}|,s為弧長,可設
\dfrac{ds}{dt}>0
\\&\implies|\alpha^\prime(t)|=\dfrac{ds}{dt},\displaystyle\int|\alpha^\prime(t)|dt=\int ds=s(t)
\\&\because\alpha是正則曲線\therefore s(t)是定義良好的,使t(s)存在
\end{aligned}$
:::
### [e.g.]1.直線
$\begin{aligned}
&\alpha(t)=\vec{a}t+\vec{b},其中\vec{a},\vec{b}是非零常數向量,且|\vec{a}|\not=1
\\&\implies s(t)=\displaystyle\int_0^t|\vec{a}|du=|\vec{a}|t,t(s)=\dfrac{s}{|\vec{a}|},\alpha(s)=\dfrac{\vec{a}}{|\vec{a}|}s+\vec{b}
\end{aligned}$
### [e.g.]2.對數螺線
$\begin{aligned}
&\alpha(t)=(ae^{bt}\cos(t),ae^{bt}\sin(t)),a>0,b<0
\\&\alpha^\prime(t)=ae^{bt}(b\cos(t)-\sin(t),b\sin(t)+\cos(t))
\\&若b\cos(t)-\sin(t)=b\sin(t)+\cos(t)=0,則有\sin^2(t)+\cos^2(t)=0(\rightarrow\leftarrow)
\\&\implies\alpha(t)是正則曲線
\\&\implies s(t)=\displaystyle\int_0^t|\alpha^\prime(u)|du=\int^t_0a\sqrt{b^2+1}e^{bu}du=\dfrac{a\sqrt{b^2+1}}{b}(e^{bt}-1)
\\&\implies t(s)=\dfrac{1}{b}\ln(1+\dfrac{bs}{a\sqrt{b^2+1}}),
\\&\alpha(s)=(a+\dfrac{bs}{\sqrt{b^2+1}})(\cos(\dfrac{1}{b}\ln(1+\dfrac{bs}{a\sqrt{b^2+1}})),\sin(\dfrac{1}{b}\ln(1+\dfrac{bs}{a\sqrt{b^2+1}})))
\end{aligned}$
## 曲率(弧長參數曲線)
$給定一個弧長參數曲線\alpha(s),\alpha^\prime(s)是$==$單位切向量$==$,記為\vec{T}$
$而\vec{T}^\prime=\alpha^{\prime\prime}(s)表示\vec{T}的變化(同時給予曲線的彎曲程度一個指標)\\\implies可定義曲率:$
:::success
$\kappa(s)\overset{def}{=}|\vec{T}'|=|\alpha''(s)|$
:::
$同時,因為\vec{T}\cdot\vec{T}=1,得到\vec{T}\cdot\vec{T}'=0,即\vec{T}\perp\vec{T}',\\可以定義單位法向量\vec{N}:$
:::success
$\vec{N}\overset{def}{=}\dfrac{\vec{T}'}{|\vec{T}'|}=\dfrac{\vec{T}'}{\kappa}(\kappa\not=0),或\vec{T}'=\kappa\vec{N}$
:::
$並可藉此定義密切平面:$
:::success
$密切平面\overset{def}{=}Span(\vec{T},\vec{N})$
:::
### [e.g.]1.直線
$\alpha(s)=\dfrac{\vec{a}}{|\vec{a}|}s+\vec{b},其中\vec{a},\vec{b}是非零常數向量,且|\vec{a}|\not=1\\\implies\kappa(s)=|\alpha''(s)|\equiv0$
:::danger
* $若\kappa(s)\equiv0,可得\alpha''(s)=0,即\alpha'(s)是常數向量,\alpha(s)可以寫成\vec{a}s+\vec{b},$$\\其中|\vec{a}|=1,\vec{a},\vec{b}為非零常數向量\\即\alpha(s)是直線\iff\kappa(s)\equiv0$
:::
### [e.g.]2.圓
$\begin{aligned}
\alpha(t)&=(a\cos(t),a\sin(t)),a>0
\\&\implies s(t)=at,t(s)=\dfrac{s}{a},\alpha(s)=(a\cos(\dfrac{s}{a}),a\sin(\dfrac{s}{a}))
\\&\implies\vec{T}=\alpha'(s)=(-\sin(\dfrac{s}{a}),\cos(\dfrac{s}{a})),\vec{T}'=(-\dfrac{1}{a}\cos(\dfrac{s}{a}),-\dfrac{1}{a}\sin(\dfrac{s}{a}))
\\&\implies\kappa(s)=\dfrac{1}{a},\vec{N}=(-\cos(\dfrac{s}{a}),-\sin(\dfrac{s}{a}))
\end{aligned}$
:::danger
* $從這個例子可看出,曲率也可以定義成曲線的密切圓半徑之倒數$
:::
## 曲率(一般曲線)
$對於一般的正則曲線\alpha(t),可將\vec{T},\vec{T}'寫成以下形式:$
:::warning
$\vec{T}=\alpha'(s)=\dfrac{\alpha'(t)}{|\alpha'(t)|}=\alpha'(t)\dfrac{dt}{ds}\implies\vec{T}'=\dfrac{\alpha''(t)}{|\alpha'(t)|^2}+\alpha'(t)\dfrac{d^2t}{ds^2}$
:::
$計算\vec{T}\cdot\vec{T}'可得:$
:::warning
$0=\vec{T}\cdot\vec{T}'=\dfrac{\alpha'(t)\cdot\alpha''(t)}{|\alpha'(t)|^3}+|\alpha'(t)|\dfrac{d^2t}{ds^2}\implies\dfrac{d^2t}{ds^2}=\dfrac{\alpha'(t)\cdot\alpha''(t)}{|\alpha'(t)|^4}$
:::
$代回\vec{T}':$
:::warning
$\vec{T}'=\dfrac{\alpha''(t)}{|\alpha'(t)|^2}+\alpha'(t)\dfrac{\alpha'(t)\cdot\alpha''(t)}{|\alpha'(t)|^4}$
:::
$由|\vec{T}'|可得曲率:$
:::warning
$\begin{aligned}
\kappa=|\vec{T}'|&=\sqrt{\dfrac{|\alpha''(t)|^2}{|\alpha'(t)|^4}+\dfrac{(\alpha'(t)\cdot\alpha''(t))^2}{|\alpha'(t)|^6}-\dfrac{2(\alpha'(t)\cdot\alpha''(t))^2}{|\alpha'(t)|^6}}
\\&=\sqrt{\dfrac{|\alpha''(t)|^2}{|\alpha'(t)|^4}-\dfrac{(\alpha'(t)\cdot\alpha''(t))^2}{|\alpha'(t)|^6}}
\\&=\dfrac{\sqrt{|\alpha''(t)|^2|\alpha'(t)|^2-(\alpha'(t)\cdot\alpha''(t))^2}}{|\alpha'(t)|^3}
\\&=\dfrac{|\alpha'(t)\wedge\alpha''(t)|}{|\alpha'(t)|^3}
\end{aligned}$
:::
$得到曲率一般公式:$
:::success
$\kappa(t)=\dfrac{|\alpha'(t)\wedge\alpha''(t)|}{|\alpha'(t)|^3}$
:::
## 扭率(弧長參數曲線)
$給定一個弧長參數曲線\alpha(s),切向量\vec{T},法向量\vec{N},\\定義副法向量\vec{B}:$
:::success
$\vec{B}\overset{def}{=}\vec{T}\wedge\vec{N}$
:::
$\begin{aligned}
&可看出|\vec{B}|=|\vec{T}||\vec{N}|\sin(\dfrac{\pi}{2})=1,且和\vec{T},\vec{N}皆垂直
\\&\implies\vec{B}\perp Span(\vec{T},\vec{N})(密切平面),\vec{B}的變化可度量曲線偏離平面(扭曲)的程度,
\\&並可以利用\vec{B}定義法平面和從切平面:
\end{aligned}$
:::success
$法平面\overset{def}{=}Span(\vec{N},\vec{B})\\從切平面\overset{def}{=}Span(\vec{B},\vec{T})$
:::
$\begin{aligned}
&而因為\vec{B}\cdot\vec{B}=1,得到\vec{B}\cdot\vec{B}'=0,又\vec{B}'\cdot\vec{T}=(\vec{T}'\wedge\vec{N}+\vec{T}\wedge\vec{N}')\cdot\vec{T}=0
\\&\implies\vec{B}'//\vec{N},即可定義扭率:
\end{aligned}$
:::success
$\tau(s)\overset{def}{=}-\vec{B}'\cdot\vec{N}=\vec{N}'\cdot\vec{B},或\vec{B}'=-\tau\vec{N}$
:::
### [e.g.]螺線
$\begin{aligned}
\alpha(s)&=(a\cos(\dfrac{s}{c}),a\sin(\dfrac{s}{c}),\dfrac{bs}{c}),其中c=\sqrt{a^2+b^2}
\\&\implies\vec{T}=(-\dfrac{a}{c}\sin(\dfrac{s}{c}),\dfrac{a}{c}\cos(\dfrac{s}{c}),\dfrac{b}{c}),|\vec{T}|=1
\\&\implies\vec{T}'=(-\dfrac{a}{c^2}\cos(\dfrac{s}{c}),-\dfrac{a}{c^2}\sin(\dfrac{s}{c}),0),\kappa(s)=\dfrac{a}{c^2}=\dfrac{a}{a^2+b^2}
\\&\implies\vec{N}=(-\cos(\dfrac{s}{c}),-\sin(\dfrac{s}{c}),0),\vec{N}'=(\dfrac{1}{c}\sin(\dfrac{s}{c}),-\dfrac{1}{c}\cos(\dfrac{s}{c}),0)
\\&\implies\tau(s)=(\vec{T}\wedge\vec{N})\cdot\vec{N}'=\begin{vmatrix}-\dfrac{a}{c}\sin(\dfrac{s}{c})&\dfrac{a}{c}\cos(\dfrac{s}{c})&\dfrac{b}{c}\\-\cos(\dfrac{s}{c})&-\sin(\dfrac{s}{c})&0\\\dfrac{1}{c}\sin(\dfrac{s}{c})&-\dfrac{1}{c}\cos(\dfrac{s}{c})&0\end{vmatrix}=\dfrac{b}{a^2+b^2}
\end{aligned}$
## 扭率(一般曲線)
$依據前面曲率的推導,可計算\vec{N}:$
:::warning
$\begin{aligned}
&\vec{T}'=\dfrac{\alpha''(t)}{|\alpha'(t)|^2}+\alpha'(t)\dfrac{\alpha'(t)\cdot\alpha''(t)}{|\alpha'(t)|^4},\kappa(t)=\dfrac{|\alpha'(t)\wedge\alpha''(t)|}{|\alpha'(t)|^3}
\\&\implies\vec{N}=\dfrac{|\alpha'(t)|\alpha''(t)}{|\alpha'(t)\wedge\alpha''(t)|}+\alpha'(t)\dfrac{\alpha'(t)\cdot\alpha''(t)}{|\alpha'(t)||\alpha'(t)\wedge\alpha''(t)|}
\end{aligned}$
:::
$可依此計算\vec{B}和\vec{B}':$
:::warning
$\begin{aligned}
\vec{B}&=\vec{T}\wedge\vec{N}=\dfrac{\alpha'(t)\wedge\alpha''(t)}{|\alpha'(t)\wedge\alpha''(t)|}
\end{aligned}$
$\begin{aligned}\implies\vec{B}'=\dfrac{|\alpha'(t)\wedge\alpha''(t)|(\alpha'(t)\wedge\alpha''(t))'-(|\alpha'(t)\wedge\alpha''(t)|)'(\alpha'(t)\wedge\alpha''(t))}{|\alpha'(t)\wedge\alpha''(t)|^2}
\\&=\dfrac{\alpha'(t)\wedge\alpha''(t)}{|\alpha'(t)\wedge\alpha''(t)|}-\dfrac{\alpha'(t)\wedge\alpha''(t)}{|\alpha'(t)\wedge\alpha''(t)|^2}(\dfrac{\alpha'(t)\wedge\alpha''(t))\cdot(\alpha'(t)\wedge\alpha'''(t)}{|\alpha'(t)\wedge\alpha''(t)|})
\end{aligned}$

:::
$由-\vec{B}'\cdot\vec{N}可得扭率:$
:::warning
$\tau=-\vec{B}'\cdot\vec{N}=-(\dfrac{\alpha'(t)\wedge\alpha'''(t)\cdot\alpha''(t)}{|\alpha'(t)\wedge\alpha''(t)|^2}+0+0+0)\\
=\dfrac{\alpha'(t)\wedge\alpha''(t)\cdot\alpha'''(t)}{|\alpha'(t)\wedge\alpha''(t)|^2}$
:::
$得到扭率一般公式:$
:::success
$\tau(t)=\dfrac{\alpha'(t)\wedge\alpha''(t)\cdot\alpha'''(t)}{|\alpha'(t)\wedge\alpha''(t)|^2}$
:::
## 弗萊納公式(Frenet formula)
$弗萊納標架(Frenet\ frame)的定義:$
:::success
$對參數弧長曲線\alpha(s),\{\vec{T},\vec{N},\vec{B}\}稱為其弗萊納標架$
:::
$而現在我們已有關於\vec{T}',\vec{B}'的表達式,可以藉此推出\vec{N}的表達式:$
:::warning
$\begin{aligned}
\begin{cases}\vec{T}'=\kappa\vec{N}\\\vec{B}'=-\tau\vec{N}\end{cases},\vec{N}'&=(\vec{B}\wedge\vec{T})'=\vec{B}'\wedge\vec{T}+\vec{B}\wedge\vec{T}'
\\&=\tau\vec{T}\wedge\vec{N}+\kappa\vec{B}\wedge\vec{N}=\tau\vec{B}-\kappa\vec{T}
\end{aligned}$
:::
$三個公式整理在一起就稱為弗萊納公式:$
:::success
$\begin{aligned}
&\begin{cases}\vec{T}'=\kappa\vec{N}\\\vec{N}'=\tau\vec{B}-\kappa\vec{T}\\\vec{B}'=-\tau\vec{N}\end{cases}或\begin{bmatrix}\vec{T}'\\\vec{N}'\\\vec{B}'\end{bmatrix}=\begin{bmatrix}O_3&\kappa I_3&O_3\\-\kappa I_3&O_3&\tau I_3\\O_3&-\tau I_3&O_3\end{bmatrix}\begin{bmatrix}\vec{T}\\\vec{N}\\\vec{B}\end{bmatrix}
\\&其中O_3=\begin{bmatrix}0&0&0\\0&0&0\\0&0&0\end{bmatrix},I_3=\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}
\end{aligned}$
:::
## 定理:平面曲線
### 定理內容
:::info
$一曲線\alpha(\kappa>0)為平面曲線\iff\tau\equiv0$
:::
### 證明
:::warning
$\begin{aligned}
(\Longrightarrow)&若\alpha(s)是平面曲線,則其密切平面固定\implies\vec{B}=常數向量
\\&\implies\vec{B}'=\vec{0},\tau\equiv0_\square
\end{aligned}$
$\begin{aligned}
(\Longleftarrow)&若\tau\equiv0,即\vec{B}是常數向量\implies(\alpha(s)\cdot\vec{B})'=\vec{T}\cdot\vec{B}+0=0
\\&\implies\alpha(s)\cdot\vec{B}(常數向量)=常數\implies\alpha(s)是平面曲線_\square
\end{aligned}$
:::
## 曲線基本定理(fundamental theorem of the curve(local))
### 剛體運動(rigid motion)
$定義:$
:::success
$\mathbb{R^3}中的剛體運動由平移變換和行列式為正的正交變換組成,$$兩者定義如下:\\\begin{cases}平移變換:A:\mathbb{R^3}\to\mathbb{R}^3,A(\vec{p})=\vec{p}+\vec{v},\vec{p},\vec{v}\in\mathbb{R}^3\\正交變換:\rho:\mathbb{R}^3\to\mathbb{R}^3\ni\rho\vec{u}\cdot\rho\vec{v}=\vec{u}\cdot\vec{v}\ \forall u,v\in\mathbb{R}^3\end{cases}$
:::
:::danger
* $\mathbb{R}^3中的剛體運動並不改變曲線的曲/扭率及弧長$
:::
### 定理內容
:::info
$\begin{aligned}
&給定可微函數\kappa(s),\tau(s),s\in I,必存在一弧長參數曲線\alpha(s)\\&使得s為\alpha(s)的弧長,
並在剛體變換之下唯一
\\&(i.e.\ \exists M:\mathbb{R}^3\to\mathbb{R}^3\ni\bar{\alpha}=M\alpha=\rho\alpha+\vec{c})
\end{aligned}$
:::
### 唯一性的證明
:::warning
$\begin{aligned}
&考慮兩弧長參數曲線\alpha(s),\bar{\alpha}(s),使兩者擁有相同的弧長及曲/扭率
\\&設s_0\in I,考慮\beta(s)=\bar{\alpha}(s)-(\bar{\alpha}(s_0)-\alpha(s_0)),即有\beta(s_0)=\alpha(s_0)
\\&\implies\beta(s)是一平移變換
\\&將\beta(s)旋轉(稱旋轉後曲線\theta(s)),使\theta'(s_0)=\alpha'(s_0),\theta''(s)=\alpha''(s)
\\&現在設\{\vec{T},\vec{N},\vec{B}\},\{\vec{\bar{T}},\vec{\bar{N}},\vec{\bar{B}}\}為\alpha,\theta的弗萊納標架,
\\&考慮函數:f(s)=|\vec{T}-\vec{\bar{T}}|^2+|\vec{N}-\vec{\bar{N}}|^2+|\vec{B}-\vec{\bar{B}}|^2
\end{aligned}$
$\begin{aligned}
有f'(s)&=2[(\vec{T}-\vec{\bar{T}})\cdot(\vec{T}'-\vec{\bar{T}}')+(\vec{N}-\vec{\bar{N}})\cdot(\vec{N}'-\vec{\bar{N}}')+(\vec{B}-\vec{\bar{B}})\cdot(\vec{B}'-\vec{\bar{B}}')]
\\&=2[\kappa(\vec{T}-\vec{\bar{T}})\cdot(\vec{N}-\vec{\bar{N}})+(\vec{N}-\vec{\bar{N}})\cdot(\vec{N}'-\vec{\bar{N}}')-\tau(\vec{B}-\vec{\bar{B}})\cdot(\vec{N}-\vec{\bar{N}})]
\\&=2(\vec{N}-\vec{\bar{N}})\cdot[\kappa(\vec{T}-\vec{\bar{T}})-\tau(\vec{B}-\vec{\bar{B}})+(\vec{N}'-\vec{\bar{N}}')]=0
\\又f(s_0)=0&\implies f(s)\equiv0,\{\vec{T},\vec{N},\vec{B}\}\equiv\{\vec{\bar{T}},\vec{\bar{N}},\vec{\bar{B}}\}
\\再考慮函數:&g(s)=|\alpha(s)-\theta(s)|^2,g'(s)=2(\alpha(s)-\theta(s))\cdot(\alpha'(s)-\theta'(s))=0
\\又g(s_0)=0&\implies g(s)\equiv0,\alpha(s)\equiv\theta(s)_\square
\end{aligned}$
:::
### 存在性的證明
:::warning
$\begin{aligned}
&要找到滿足的曲線,等同在給定\kappa,\tau,s的情況下找到弗萊納公式的解:
\\&\begin{bmatrix}\vec{T}'\\\vec{N}'\\\vec{B}'\end{bmatrix}=\underbrace{\begin{bmatrix}O_3&\kappa I_3&O_3\\-\kappa I_3&O_3&\tau I_3\\O_3&-\tau I_3&O_3\end{bmatrix}}_{A(s)}\underbrace{\begin{bmatrix}\vec{T}\\\vec{N}\\\vec{B}\end{bmatrix}}_{h(s)}(9\wedge9的線性常微分方程式)
\\&可取初始值\vec{T}(s_0)=\vec{e_1},\vec{N}=\vec{e_2},\vec{B}=\vec{e_3}使\begin{cases}h'(s)=A(s)h(s)\\h(s_0)=(\vec{e_1},\vec{e_2},\vec{e_3})\end{cases}有唯一解
\end{aligned}$
:::
:::warning
$\begin{aligned}
&接下來證明\{\vec{T},\vec{N},\vec{B}\}構成單範正交基底(長度1,彼此垂直):
\\&\begin{cases}(|\vec{T}|^2)'=2\kappa\vec{T}\cdot\vec{N}\\(|\vec{N}|^2)'=2\tau\vec{T}\cdot\vec{N}-2\kappa\vec{B}\cdot\vec{N}\\(|\vec{T}|^2)'=-2\tau\vec{T}\cdot\vec{N}\end{cases},\begin{cases}(\vec{T}\cdot\vec{N})'=\kappa|\vec{N}|^2-\kappa|T|^2-\tau\vec{T\cdot}\vec{B}\\(\vec{N}\cdot\vec{B})'=\tau|\vec{N}|^2+\tau|B|^2-\kappa\vec{T\cdot}\vec{B}\\(\vec{T}\cdot\vec{B})'=\kappa\vec{B}\cdot\vec{N}-\tau\vec{T\cdot}\vec{N}\end{cases}
\\&令l(s)=(|\vec{T}|^2,|\vec{N}|^2,|\vec{B}|^2,\vec{T}\cdot\vec{N},\vec{N}\cdot\vec{B},\vec{T}\cdot\vec{B}),即有線性常微分方程式:
\\&\begin{cases}l'(s)=M(s)l(s)\\l(s_0)=(1,1,1,0,0,0)\end{cases},M=\begin{bmatrix}0&0&0&2\kappa&0&0\\0&0&0&2\tau&-2\kappa&0\\0&0&0&0&-2\tau&0\\-\kappa&\kappa&0&0&0&-\tau\\0&\tau&\tau&0&0&-\kappa\\0&0&0&-\tau&\kappa&0\end{bmatrix}
\\&又(1,1,1,0,0,0)是一組解(也是唯一一組),得l(s)=(1,1,1,0,0,0)_\square
\end{aligned}$
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# (III) 等周不等式
## 封閉曲線
$封閉曲線的定義:$
:::success
$平面曲線\alpha:I\to\mathbb{R}^3是封閉曲線\iff對端點a,b,\alpha^{(n)}(a)=\alpha^{(n)}(b)$
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$同時也可定義簡單封閉曲線:$
:::success
$沒有自交點的封閉曲線稱為簡單封閉曲線,\\如下圖C是簡單封閉曲線,C'則不是$
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## 等周不等式(isoperimetric inequality)
### 定理內容
:::info
$\begin{aligned}
&給定固定長度\ell的簡單封閉曲線,假設內部圍成面積為A,
\\&則有以下不等式:A\le\dfrac{\ell^2}{4\pi},等號成立於該曲線為圓
\end{aligned}$
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### 證明
:::warning
$\begin{aligned}
&利用格林定理:\displaystyle\oint_{C^+}Pdx+Qdy=\displaystyle\iint_R(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y})dxdy
\\&P=-y(t),Q=x(t)代入,得\displaystyle\oint_{C^+}(x'y-xy')dt=\displaystyle2\iint_Rdxdy=2Area(R)
\\&\implies Area(R)=\dfrac{1}{2}\displaystyle\oint_{C^+}(x'y-xy')dt
\end{aligned}$
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:::warning
$\begin{aligned}
&如圖,假設兩弧長參數曲線\alpha(s),C(s)(圓),s\in[0,\ell]
\\&\because兩者對齊\therefore可設\alpha(s)=(x(s),y(s)),C(s)=(x(s),\bar{y}(s))
\\&又\displaystyle\int_0^\ell x'yds=xy\bigg|^\ell_0-\int_0^\ell xy'ds=-\int_0^\ell xy'ds\end{aligned}$
$\begin{aligned}
\implies A+\pi r^2&=\dfrac{1}{2}\displaystyle\oint_{C^+}(x'\bar{y}-x\bar{y}')ds+\dfrac{1}{2}\displaystyle\oint_{\alpha^+}(x'y-xy')ds
\\&=\dfrac{1}{2}\displaystyle\int_0^\ell2(x'\bar{y}-xy')ds=\displaystyle\int_0^\ell(x'\bar{y}-xy')ds
\\&\le\displaystyle\int_0^\ell\sqrt{(x'\bar{y}-xy')^2}ds
\\&\le\displaystyle\int_0^\ell\sqrt{(x'^2+y'^2)(x^2+\bar{y}^2)}ds=\ell r\end{aligned}$
$\begin{aligned}
&\implies A+\pi r^2\le\ell r,又A+\pi r^2\ge2\sqrt{A\pi r^2}
\\&\implies2\sqrt{A\pi r^2}\le\ell r,A\le\dfrac{\ell^2}{4\pi}_\square
\end{aligned}$
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:::danger
* $分段連續(即有有限個尖點)的曲線同樣滿足此不等式$
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