圓錐坐標系

圓錐坐標系（英語：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面，錐軸為 x-軸的圓錐面，錐軸為 z-軸的圓錐面。

基本定義

圓錐坐標 $(r,\ \mu ,\ \nu )$ 通常定義為

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

$r$ 坐標曲面是圓心在原點的圓球面：

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

。

$\mu$ 與 $\nu$ 坐標曲面是兩個相交的圓錐面：

{\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

。

如同球坐標系，徑向坐標 $r$ 的標度因子是

h_{r}=1

。

坐標 $\mu$ 與 $\nu$ 的標度因子分別是

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)}}}

。

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