查看“圓錐坐標系”的源代码

[[File:Conical_coordinates.png|thumb|380px|right|圓錐坐標系的幾個[[坐標曲面]]。紅色圓球面的 <math>r=2</math> 。錐軸為 z-軸的藍色圓錐面的 <math>\mu=\cosh(1)</math> 。錐軸為 x-軸的黃色圓錐面 <math>\nu^2= 2/3</math> 。z-軸是垂直的，以白色表示。 x-軸以綠色表示。三個坐標曲面相交於點 P （以黑色的圓球表示），[[直角坐標]]大約為 <math>(1.26,\  - 0.78,\  1.34)</math> 。]]

'''圓錐坐標系'''（{{lang-en|Conical coordinates}}）是一種三維[[正交坐標系]]。它的三個[[坐標曲面]]分別為同心[[圓球|圓球面]]，錐軸為 x-軸的[[圓錐|圓錐面]]，錐軸為 z-軸的圓錐面。

==基本定義==
圓錐坐標 <math>(r,\ \mu,\ \nu)</math> 通常定義為
:<math>x = \frac{r\mu\nu}{bc}</math> 、
:<math>y = \frac{r}{b} \sqrt{\frac{\left( \mu^{2} - b^{2} \right) \left( \nu^{2} - b^{2} \right)}{\left( b^{2} - c^{2} \right)} }</math> 、
:<math>z = \frac{r}{c} \sqrt{\frac{\left( \mu^{2} - c^{2} \right) \left( \nu^{2} - c^{2} \right)}{\left( c^{2} - b^{2} \right)} }</math> ；

其中，<math>0\le r < \inf </math> ，<math>\nu^{2} < c^{2} < \mu^{2} < b^{2} </math> 。

==坐標曲面==
<math>r</math> [[坐標曲面]]是圓心在原點的圓球面：
:<math>x^{2} + y^{2} + z^{2} = r^{2}</math> 。

<math>\mu</math> 與 <math>\nu</math> [[坐標曲面]]是兩個相交的圓錐面：
:<math>\frac{x^{2}}{\mu^{2}} + \frac{y^{2}}{\mu^{2} - b^{2}} + \frac{z^{2}}{\mu^{2} - c^{2}} = 0</math> 、
:<math>\frac{x^{2}}{\nu^{2}} + \frac{y^{2}}{\nu^{2} - b^{2}} + \frac{z^{2}}{\nu^{2} - c^{2}} = 0</math> 。

===標度因子===
如同[[球坐標系]]，徑向坐標 <math>r</math> 的標度因子是
:<math>h_{r} = 1</math> 。

坐標<math>\mu</math> 與 <math>\nu</math> 的標度因子分別是
:<math>h_{\mu} = r \sqrt{\frac{\mu^{2} - \nu^{2}}{\left( b^{2} - \mu^{2} \right) \left( \mu^{2} - c^{2} \right)}}</math> 、
:<math>h_{\nu} = r \sqrt{\frac{\mu^{2} - \nu^{2}}{\left( b^{2} - \nu^{2} \right) \left( c^{2} - \nu^{2} \right)}}</math> 。

==參閱==
{{正交坐標系}}

==參考目錄==
* {{cite book | author = Morse PM, Feshbach H | date = 1953 | title = Methods of Theoretical Physics, Part I | publisher = McGraw-Hill | location = New York | id = ISBN 0-07-043316-X| pages = p. 659}}
* {{cite book | author = Margenau H, Murphy GM | year = 1956 | title = The Mathematics of Physics and Chemistry | publisher = D. van Nostrand | location = New York| pages = pp. 183–184 }}
* {{cite book | author = Korn GA, Korn TM |date = 1961 | title = Mathematical Handbook for Scientists and Engineers | publisher = McGraw-Hill | location = New York | id = ASIN B0000CKZX7 | pages = p. 179}}
* {{cite book | author = Sauer R, Szabó I | date = 1967 | title = Mathematische Hilfsmittel des Ingenieurs | publisher = Springer Verlag | location = New York| pages = pp. 991–100}}  
* {{cite book | author = Arfken G | date = 1970 | title = Mathematical Methods for Physicists | edition = 2nd ed. | publisher = Academic Press | location = Orlando, FL | pages = pp. 118-119 | id = ASIN B000MBRNX4}}
* {{cite book | author = Moon P, Spencer DE | date = 1988 | chapter = Conical Coordinates (r, θ, λ) | title = Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions | edition = corrected 2nd ed., 3rd print ed. | publisher = Springer-Verlag | location = New York | pages = pp. 37-40 (Table 1.09) | isbn = 978-0387184302}}

[[Category:坐標系|Y]]