# 椭圆本文重定向自 椭圆

## 概述

${\displaystyle Ax^{2}+Bxy+Cy^{2}+Dx+Ey+F=0\,}$

## 离心率

${\displaystyle \varepsilon ={\sqrt {1-{\frac {b^{2}}{a^{2}}}}}}$

${\displaystyle \varepsilon ={\frac {c}{a}}}$

## 方程

${\displaystyle {\frac {(x-h)^{2}}{a^{2}}}+{\frac {(y-k)^{2}}{b^{2}}}=1}$

${\displaystyle x=h+a\,\cos t,\,\!}$
${\displaystyle y=k+b\,\sin t\,\!}$

${\displaystyle x=a\,\cos t,\,\!}$
${\displaystyle y=b\,\sin t\,\!}$

 椭圆方程 ${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1(a>b>0)}$ ${\displaystyle {\frac {y^{2}}{a^{2}}}+{\frac {x^{2}}{b^{2}}}=1(a>b>0)}$ 图像 范围 ${\displaystyle -a\leq x\leq a,-b\leq y\leq b}$ ${\displaystyle -a\leq y\leq a,-b\leq x\leq b}$

### 相对于中心的极坐标形式

${\displaystyle {\overline {CP}}=r'={\frac {ab}{\sqrt {a^{2}\sin ^{2}\psi +b^{2}\cos ^{2}\psi }}}={\frac {b}{\sqrt {1-\varepsilon ^{2}\cos ^{2}\psi }}}}$

### 相对于焦点的极坐标形式

${\displaystyle {\overline {F_{1}P}}=r={\frac {a\cdot (1-\varepsilon ^{2})}{1-\varepsilon \cdot \cos \theta }}}$

#### 半正焦弦和极坐标

${\displaystyle r\cdot (1+\varepsilon \cdot \cos \theta )=\ell \,\!}$

## 面积和周长

${\displaystyle C=2\pi a\left[{1-\left({1 \over 2}\right)^{2}{\frac {c^{2}}{a^{2}}}-\left({1\cdot 3 \over 2\cdot 4}\right)^{2}{c^{4} \over {3a^{4}}}-\left({1\cdot 3\cdot 5 \over 2\cdot 4\cdot 6}\right)^{2}{c^{6} \over {5a^{6}}}-\dots }\right]\!\,}$

${\displaystyle C=-2\pi a\sum _{n=0}^{\infty }{\left\lbrace \left[\prod _{m=1}^{n}\left({2m-1 \over 2m}\right)\right]^{2}{c^{2n} \over {{a^{2n}}\left(2n-1\right)}}\right\rbrace }}$

${\displaystyle C\approx \pi \left[3(a+b)-{\sqrt {(3a+b)(a+3b)}}\right]\!\,}$

${\displaystyle C\approx 3a\pi \left[1+{\sqrt {1-\left({\frac {c}{a}}\right)^{2}}}\right]-a\pi {\sqrt {\left[3+{\sqrt {1-\left({\frac {c}{a}}\right)^{2}}}\right]\left[1+3{\sqrt {1-\left({\frac {c}{a}}\right)^{2}}}\right]}}\!\,}$

${\displaystyle C\approx \pi (a+b)\left[1+{\frac {3\left({\frac {a-b}{a+b}}\right)^{2}}{10+{\sqrt {4-3\left({\frac {a-b}{a+b}}\right)^{2}}}}}\right]\left[1+\left({\frac {22}{7\pi }}-1\right)\left({\frac {a-b}{a}}\right)^{33}{\sqrt[{1000}]{\left({\frac {a-b}{a}}\right)^{697}}}\right]\!\,}$

## 标准方程的推导

• 如果在一个平面内一个动点到两个定点距离的和等于定长，那么这个动点的轨迹叫做椭圆。

${\displaystyle |PF_{1}|+|PF_{2}|=2a(a>0)\,}$，其中${\displaystyle 2a\,}$为定长。

${\displaystyle {\sqrt {(x+c)^{2}+y^{2}}}=2a-{\sqrt {(x-c)^{2}+y^{2}}}\,}$

${\displaystyle (a^{2}-c^{2})x^{2}+a^{2}y^{2}=a^{2}(a^{2}-c^{2})\,}$

${\displaystyle a>c\,}$时，并设${\displaystyle a^{2}-c^{2}=b^{2}\,}$，则①式可以进一步化简：

${\displaystyle b^{2}x^{2}+a^{2}y^{2}=a^{2}b^{2}\,}$

${\displaystyle {\frac {x^{2}}{a^{2}}}+{\frac {y^{2}}{b^{2}}}=1\,}$

• 椭圆的图像如果在直角坐标系中表示，那么上述定义中两个定点被定义在了x轴。若将两个定点改在y轴，可以用相同方法求出另一个椭圆的标准方程
${\displaystyle {\frac {y^{2}}{a^{2}}}+{\frac {x^{2}}{b^{2}}}=1(a>b>0)\,}$
• 在方程中，所设的${\displaystyle 2a\,}$称为长轴长，${\displaystyle 2b\,}$称为短轴长，而所设的定点称为焦点，那么${\displaystyle 2c\,}$称为焦距。在假设的过程中，假设了${\displaystyle a>c\,}$，如果不这样假设，会发现得不到椭圆。当${\displaystyle a=c\,}$时，这个动点的轨迹是一个线段；当${\displaystyle a时，根本得不到实际存在的轨迹，而这时，其轨迹称为虚椭圆。另外还要注意，在假设中，还有一处：${\displaystyle a^{2}-c^{2}=b^{2}\,}$
• 通常认为是椭圆的一种特殊情况。

## 椭圆的旋转和平移

${\displaystyle A(x-u)^{2}+2B(x-u)(y-v)+C(y-v)^{2}+f=0\,}$

${\displaystyle x=x^{\prime }-u}$
${\displaystyle y=y^{\prime }-v}$

${\displaystyle B\neq 0}$，则表示椭圆的长短轴与坐标系的坐标轴并不平行或垂直，即发生了旋转。设旋转的角度为${\displaystyle \displaystyle \varphi }$，则有

${\displaystyle \displaystyle tan(2\varphi )={\frac {B}{A-C}}}$

${\displaystyle A-C=0}$，则说明${\displaystyle \varphi =\pm {\frac {\pi }{4}}}$

${\displaystyle x=x^{\prime }\cos \varphi -y^{\prime }\sin \varphi }$
${\displaystyle y=y^{\prime }\cos \varphi +x^{\prime }\sin \varphi }$

## 渐开线及其导数

${\displaystyle {\begin{cases}x=a\cos t+{\cfrac {abE\left(t,{\cfrac {\sqrt {a^{2}-b^{2}}}{a}}\right)\sin t}{\sqrt {a^{2}\sin ^{2}t+b^{2}\cos ^{2}t}}}\!\,\\\\y=b\sin t+{\cfrac {b^{2}E\left(t,{\cfrac {\sqrt {a^{2}-b^{2}}}{a}}\right)\cos t}{\sqrt {a^{2}\sin ^{2}t+b^{2}\cos ^{2}t}}}\!\,\\\end{cases}}}$

${\displaystyle {\begin{cases}{\cfrac {{\rm {d}}x}{\rm {{d}t}}}={\cfrac {\left[b^{2}\sin 2t-2b^{2}\sin t\cdot E\left(t,{\cfrac {\sqrt {a^{2}-b^{2}}}{a}}\right)\right]\left(a^{2}\sin ^{2}t+b^{2}\cos ^{2}t\right)-ab\left(a^{2}-b^{2}\right)\sin 2t\cdot E\left(t,{\cfrac {\sqrt {a^{2}-b^{2}}}{a}}\right)\sin t}{2\left(a^{2}\sin ^{2}t+b^{2}\cos ^{2}t\right){\sqrt {a^{2}\sin ^{2}t+b^{2}\cos ^{2}t}}}}-a\sin t\!\,\\\\{\cfrac {{\rm {d}}y}{\rm {{d}t}}}={\cfrac {\left[b^{3}\sin 2t-2ab^{2}\sin t\cdot E\left(t,{\cfrac {\sqrt {a^{2}-b^{2}}}{a}}\right)\right]\left(a^{2}\sin ^{2}t+b^{2}\cos ^{2}t\right)-ab^{2}\left(a^{2}-b^{2}\right)\sin 2t\cdot E\left(t,{\cfrac {\sqrt {a^{2}-b^{2}}}{a}}\right)\sin t}{2a\left(a^{2}\sin ^{2}t+b^{2}\cos ^{2}t\right){\sqrt {a^{2}\sin ^{2}t+b^{2}\cos ^{2}t}}}}+b\cos t\!\,\\\end{cases}}}$