橢球坐標系

橢球坐標系（英語：Ellipsoidal coordinates）是一種三維正交坐標系，是橢圓坐標系的推廣。與大多數的三維正交坐標系的生成方法不同，橢球坐標系不是由任何二維正交坐標系延伸或旋轉生成的。

橢球坐標系: a=1,b=0.8,c=0.6,
λ=-0.1 紅色橢球, μ=-0.5 藍色單葉雙曲面, ν=-0.8 品紅色雙葉雙曲面.

基本公式

橢球坐標 ${\displaystyle (\lambda ,\ \mu ,\ \nu )}$ 以直角坐標 ${\displaystyle (x,\ y,\ z)}$ 定義為：

${\displaystyle x^{2}={\frac {(a^{2}+\lambda )(a^{2}+\mu )(a^{2}+\nu )}{(a^{2}-b^{2})(a^{2}-c^{2})}}}$ 、

${\displaystyle y^{2}={\frac {(b^{2}+\lambda )(b^{2}+\mu )(b^{2}+\nu )}{(b^{2}-a^{2})(b^{2}-c^{2})}}}$ 、

${\displaystyle z^{2}={\frac {(c^{2}+\lambda )(c^{2}+\mu )(c^{2}+\nu )}{(c^{2}-b^{2})(c^{2}-a^{2})}}}$ ；

其中，橢球坐標遵守以下限制：

${\displaystyle -\lambda <c^{2}<-\mu <b^{2}<-\nu <a^{2}}$ 。

坐標曲面

橢球上的與雙曲面相交的曲線，a=1, b=0.8, c=0.6.

${\displaystyle \lambda }$-坐標曲面是橢球面 ：

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

${\displaystyle \mu }$-坐標曲面是單葉雙曲面 (hyperboloid of one sheet) ：

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

${\displaystyle \nu }$-坐標曲面是雙葉雙曲面 (hyperboloid of two sheet) ：

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

標度因子

為了簡化標度因子的計算，設定函數

${\displaystyle S(\sigma )\ {\stackrel {\mathrm {def} }{=}}\ (a^{2}+\sigma )(b^{2}+\sigma )(c^{2}+\sigma )}$ ；

其中，參數 ${\displaystyle \sigma }$ 可以代表任何一個橢球坐標 ${\displaystyle (\lambda ,\ \mu ,\ \nu )}$ 。

橢球坐標的標度因子分別為

${\displaystyle h_{\lambda }={\frac {1}{2}}{\sqrt {\frac {(\lambda -\mu )(\lambda -\nu )}{S(\lambda )}}}}$ 、

${\displaystyle h_{\mu }={\frac {1}{2}}{\sqrt {\frac {(\mu -\lambda )(\mu -\nu )}{S(\mu )}}}}$ 、

${\displaystyle h_{\nu }={\frac {1}{2}}{\sqrt {\frac {(\nu -\lambda )(\nu -\mu )}{S(\nu )}}}}$ 。

無窮小體積元素等於

${\displaystyle dV={\frac {(\lambda -\mu )(\lambda -\nu )(\mu -\nu )}{8{\sqrt {-S(\lambda )S(\mu )S(\nu )}}}}\ d\lambda d\mu d\nu }$ 。

拉普拉斯算子是

${\displaystyle \nabla ^{2}\Phi ={\frac {4{\sqrt {S(\lambda )}}}{\left(\lambda -\mu \right)\left(\lambda -\nu \right)}}{\frac {\partial }{\partial \lambda }}\left[{\sqrt {S(\lambda )}}{\frac {\partial \Phi }{\partial \lambda }}\right]\ +\ {\frac {4{\sqrt {S(\mu )}}}{\left(\mu -\lambda \right)\left(\mu -\nu \right)}}{\frac {\partial }{\partial \mu }}\left[{\sqrt {S(\mu )}}{\frac {\partial \Phi }{\partial \mu }}\right]}$

${\displaystyle +\ {\frac {4{\sqrt {S(\nu )}}}{\left(\nu -\lambda \right)\left(\nu -\mu \right)}}{\frac {\partial }{\partial \nu }}\left[{\sqrt {S(\nu )}}{\frac {\partial \Phi }{\partial \nu }}\right]}$ 。

其它微分算子，例如 ${\displaystyle \nabla \cdot \mathbf {F} }$ 與 ${\displaystyle \nabla \times \mathbf {F} }$ ，都可以用橢球坐標表達，只需要將標度因子代入正交坐標條目內對應的一般公式。

參閱

• 橢球
• 類球面

參考目錄

• Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 663. 引文格式1維護：冗餘文本 (link)
• Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 114. ISBN 0-86720-293-9. 引文格式1維護：冗餘文本 (link)
• Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: pp. 101–102. 引文格式1維護：冗餘文本 (link)
• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 176. 引文格式1維護：冗餘文本 (link)
• Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 178–180. 引文格式1維護：冗餘文本 (link)
• Moon PH, Spencer DE. Ellipsoidal Coordinates (η, θ, λ). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions corrected 2nd ed., 3rd print ed. New York: Springer Verlag. 1988: pp. 40–44 (Table 1.10). ISBN 0-387-02732-7. 引文格式1維護：冗餘文本 (link)