椭球

（重定向自橢球體

标准方程

${\displaystyle {x^{2} \over a^{2}}+{y^{2} \over b^{2}}+{z^{2} \over c^{2}}=1}$

• ${\displaystyle a=b=c\,\!}$
• ${\displaystyle a=b>c\,\!}$ 扁球面（类似盤状）；
• ${\displaystyle a=b 长球面（类似條状）；
• ${\displaystyle a\neq b,b\neq c,c\neq a\!}$ 不等边椭球（“三条边都不相等”）。

参数化

{\displaystyle {\begin{aligned}x&=a\,\sin \theta \cos \varphi ;\!{\color {white}|}\\y&=b\,\sin \theta \sin \varphi ;\\z&=c\,\cos \theta ;\end{aligned}}\,\!}
${\displaystyle {\begin{matrix}0\leq \theta \leq {180}^{\circ };\quad {0}\leq \varphi \leq {360}^{\circ };\!{\color {white}{\big |}}\end{matrix}}\,\!}$

{\displaystyle {\begin{aligned}x&=a\,\cos \beta \cos \lambda ;\!{\color {white}|}\\y&=b\,\cos \beta \sin \lambda ;\\z&=c\,\sin \beta ;\end{aligned}}\,\!}
${\displaystyle {\begin{matrix}-90^{\circ }\leq \beta \leq 90^{\circ };\quad -180^{\circ }\leq \lambda \leq 180^{\circ };\!{\color {white}{\big |}}\end{matrix}}\,\!}$
（注意，当${\displaystyle \scriptstyle {{\color {white}|}\beta =\pm {90}^{\circ }}{\color {white}|}\,\!}$ 时，也就是在极点时，这个参数不是一一对应的）

体积和表面积

体积

${\displaystyle {\frac {4}{3}}\pi abc.\,\!}$

表面积

${\displaystyle S=2\pi \left[c^{2}+b{\sqrt {a^{2}-c^{2}}}F\left(o\!\varepsilon ,{\frac {b^{2}-c^{2}}{b^{2}\sin ^{2}o\!\varepsilon }}\right)+{\frac {bc^{2}}{\sqrt {a^{2}-c^{2}}}}E\left(o\!\varepsilon ,{\frac {b^{2}-c^{2}}{b^{2}\sin ^{2}o\!\varepsilon }}\right)\right],\,\!}$

${\displaystyle o\!\varepsilon =\arccos {\frac {c}{a}}\;}$ （扁球面）或${\displaystyle \arccos {\frac {a}{c}}\;}$ （长球面），是角离心率${\displaystyle F(x,k)\,\!}$ ${\displaystyle E(x,k)\,\!}$ 是第一类和第二类不完全椭圆积分

${\displaystyle S\approx 4\pi \!\left({\frac {a^{p}b^{p}+a^{p}c^{p}+b^{p}c^{p}}{3}}\right)^{\frac {1}{p}}.\,\!}$

扁球面：${\displaystyle S=2\pi \!\left(a^{2}+c^{2}{\frac {\operatorname {arctanh} \sin o\!\varepsilon }{\sin o\!\varepsilon }}\right);\,\!}$

${\displaystyle c\,}$ ${\displaystyle a\,}$ ${\displaystyle b\,}$ 都小很多时，表面积近似等于${\displaystyle 2\pi ab.\,\!}$

椭球与平面相交的横截面

${\displaystyle {d_{1,2}^{2}}={{8(1-{z_{c}^{2} \over {\sum _{i=1}^{3}r_{i}^{2}\sin ^{2}p_{i}}})} \over {\sum _{i=1}^{3}{\cos ^{2}p_{i} \over {r_{i}^{2}}}}\pm {\sqrt {(\sum _{i=1}^{3}{\cos ^{2}p_{i} \over {r_{i}^{2}}})^{2}-4(\sum _{i=1}^{3}r_{i}^{2}\sin ^{2}p_{i})/r_{1}^{2}r_{2}^{2}r_{3}^{2}}}}}$

质量性质

${\displaystyle m=\rho V=\rho {\frac {4}{3}}\pi abc\,\!}$

${\displaystyle I_{\mathrm {xx} }=m{b^{2}+c^{2} \over 5}}$
${\displaystyle I_{\mathrm {yy} }=m{c^{2}+a^{2} \over 5}}$
${\displaystyle I_{\mathrm {zz} }=m{a^{2}+b^{2} \over 5}}$

${\displaystyle a={\sqrt {{5 \over 2}{I_{\mathrm {yy} }+I_{\mathrm {zz} }-I_{\mathrm {xx} } \over m}}}}$
${\displaystyle b={\sqrt {{5 \over 2}{I_{\mathrm {zz} }+I_{\mathrm {xx} }-I_{\mathrm {yy} } \over m}}}}$
${\displaystyle c={\sqrt {{5 \over 2}{I_{\mathrm {xx} }+I_{\mathrm {yy} }-I_{\mathrm {zz} } \over m}}}}$
${\displaystyle \rho ={\frac {3}{4}}{m \over \pi abc}\!}$

引用

1. ^ Cayley, A. On the geodesic lines on an oblate spheroid. Phil. Mag. 1870, 40 (4th ser.): 329–340.
2. ^ Wu, Jianguo. Inferring 3D Ellipsoids based on Cross-Sectional Images with Applications to Porosity Control of Additive Manufacturing. IISE Transactions. 2018.
3. ^ Egg Curves by Jürgen Köller.