圆锥坐标系

圆锥坐标系（英语：Conical coordinates）是一种三维正交坐标系。它的三个坐标曲面分别为同心圆球面，锥轴为 x-轴的圆锥面，锥轴为 z-轴的圆锥面。

基本定义

圆锥坐标 $(r,\ \mu ,\ \nu )$ 通常定义为

x={\frac {r\mu \nu }{bc}}

、

y={\frac {r}{b}}{\sqrt {\frac {\left(\mu ^{2}-b^{2}\right)\left(\nu ^{2}-b^{2}\right)}{\left(b^{2}-c^{2}\right)}}}

、

z={\frac {r}{c}}{\sqrt {\frac {\left(\mu ^{2}-c^{2}\right)\left(\nu ^{2}-c^{2}\right)}{\left(c^{2}-b^{2}\right)}}}

；

其中， $0\leq r<\inf$ ， $\nu ^{2}<c^{2}<\mu ^{2}<b^{2}$ 。

$r$ 坐标曲面是圆心在原点的圆球面：

x^{2}+y^{2}+z^{2}=r^{2}

。

$\mu$ 与 $\nu$ 坐标曲面是两个相交的圆锥面：

{\frac {x^{2}}{\mu ^{2}}}+{\frac {y^{2}}{\mu ^{2}-b^{2}}}+{\frac {z^{2}}{\mu ^{2}-c^{2}}}=0

、

{\frac {x^{2}}{\nu ^{2}}}+{\frac {y^{2}}{\nu ^{2}-b^{2}}}+{\frac {z^{2}}{\nu ^{2}-c^{2}}}=0

。

如同球坐标系，径向坐标 $r$ 的标度因子是

h_{r}=1

。

坐标 $\mu$ 与 $\nu$ 的标度因子分别是

h_{\mu }=r{\sqrt {\frac {\mu ^{2}-\nu ^{2}}{\left(b^{2}-\mu ^{2}\right)\left(\mu ^{2}-c^{2}\right)}}}

、

h_{\nu }=r{\sqrt {\frac {\mu ^{2}-\nu ^{2}}{\left(b^{2}-\nu ^{2}\right)\left(c^{2}-\nu ^{2}\right)}}}

。

Morse PM, Feshbach H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill. 1953: p. 659. ISBN 0-07-043316-X.
Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 183–184.
Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 179. ASIN B0000CKZX7.
Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: pp. 991–100.
Arfken G. Mathematical Methods for Physicists 2nd ed. Orlando, FL: Academic Press. 1970: pp. 118–119. ASIN B000MBRNX4.
Moon P, Spencer DE. Conical Coordinates (r, 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. 37–40 (Table 1.09). ISBN 978-0387184302.