直交曲線座標
直交曲線座標の表

通常の直交曲線座標の他に、いくつかのやや珍しい直交曲線座標を以下に表に示す。 Interval notation is used for compactness in the coordinates column.
| Curvillinear coordinates (q1, q2, q3) | Transformation from cartesian (x, y, z) | Scale factors |
| --- | --- | --- |
| Spherical polar coordinates  (r,\theta ,\phi )\in [0,\infty )\times [0,\pi ]\times [0,2\pi ) |  {\begin{aligned}x&=r\sin \theta \cos \phi \\y&=r\sin \theta \sin \phi \\z&=r\cos \theta \end{aligned}} |  {\begin{aligned}h_{1}&=1\\h_{2}&=r\\h_{3}&=r\sin \theta \end{aligned}} |
| Cylindrical polar coordinates  (r,\phi ,z)\in [0,\infty )\times [0,2\pi )\times (-\infty ,\infty ) |  {\begin{aligned}x&=r\cos \phi \\y&=r\sin \phi \\z&=z\end{aligned}} |  {\begin{aligned}h_{1}&=h_{3}=1\\h_{2}&=r\end{aligned}} |
| Parabolic cylindrical coordinates  (u,v,z)\in (-\infty ,\infty )\times [0,\infty )\times (-\infty ,\infty ) |  {\begin{aligned}x&={\frac {1}{2}}(u^{2}-v^{2})\\y&=uv\\z&=z\end{aligned}} |  {\begin{aligned}h_{1}&=h_{2}={\sqrt {u^{2}+v^{2}}}\\h_{3}&=1\end{aligned}} |
| Parabolic coordinates  (u,v,\phi )\in [0,\infty )\times [0,\infty )\times [0,2\pi ) |  {\begin{aligned}x&=uv\cos \phi \\y&=uv\sin \phi \\z&={\frac {1}{2}}(u^{2}-v^{2})\end{aligned}} |  {\begin{aligned}h_{1}&=h_{2}={\sqrt {u^{2}+v^{2}}}\\h_{3}&=uv\end{aligned}} |
| Paraboloidal coordinates  {\begin{aligned}&(\lambda ,\mu ,\nu )\in [0,b^{2})\times (b^{2},a^{2})\times (a^{2},\infty )\\&b^{2}<a^{2}\end{aligned}} |  {\frac {x^{2}}{q_{i}-a^{2}}}+{\frac {y^{2}}{q_{i}-b^{2}}}=2z+q_{i} where   (q_{1},q_{2},q_{3})=(\lambda ,\mu ,\nu ) |  h_{i}={\frac {1}{2}}{\sqrt {\frac {(q_{j}-q_{i})(q_{k}-q_{i})}{(a^{2}-q_{i})(b^{2}-q_{i})}}} |
| Ellipsoidal coordinates  {\begin{aligned}&(\lambda ,\mu ,\nu )\in [0,c^{2})\times (c^{2},b^{2})\times (b^{2},a^{2})\\&\lambda <c^{2}<b^{2}<a^{2},\\&c^{2}<\mu <b^{2}<a^{2},\\&c^{2}<b^{2}<\nu <a^{2},\end{aligned}} |  {\frac {x^{2}}{a^{2}-q_{i}}}+{\frac {y^{2}}{b^{2}-q_{i}}}+{\frac {z^{2}}{c^{2}-q_{i}}}=1 where   (q_{1},q_{2},q_{3})=(\lambda ,\mu ,\nu ) |  h_{i}={\frac {1}{2}}{\sqrt {\frac {(q_{j}-q_{i})(q_{k}-q_{i})}{(a^{2}-q_{i})(b^{2}-q_{i})(c^{2}-q_{i})}}} |
| Elliptic cylindrical coordinates  (u,v,z)\in [0,\infty )\times [0,2\pi )\times (-\infty ,\infty ) |  {\begin{aligned}x&=a\cosh u\cos v\\y&=a\sinh u\sin v\\z&=z\end{aligned}} |  {\begin{aligned}h_{1}&=h_{2}=a{\sqrt {\sinh ^{2}u+\sin ^{2}v}}\\h_{3}&=1\end{aligned}} |
| Prolate spheroidal coordinates  (\xi ,\eta ,\phi )\in [0,\infty )\times [0,\pi ]\times [0,2\pi ) |  {\begin{aligned}x&=a\sinh \xi \sin \eta \cos \phi \\y&=a\sinh \xi \sin \eta \sin \phi \\z&=a\cosh \xi \cos \eta \end{aligned}} |  {\begin{aligned}h_{1}&=h_{2}=a{\sqrt {\sinh ^{2}\xi +\sin ^{2}\eta }}\\h_{3}&=a\sinh \xi \sin \eta \end{aligned}} |
| Oblate spheroidal coordinates  (\xi ,\eta ,\phi )\in [0,\infty )\times \left[-{\frac {\pi }{2}},{\frac {\pi }{2}}\right]\times [0,2\pi ) |  {\begin{aligned}x&=a\cosh \xi \cos \eta \cos \phi \\y&=a\cosh \xi \cos \eta \sin \phi \\z&=a\sinh \xi \sin \eta \end{aligned}} |  {\begin{aligned}h_{1}&=h_{2}=a{\sqrt {\sinh ^{2}\xi +\sin ^{2}\eta }}\\h_{3}&=a\cosh \xi \cos \eta \end{aligned}} |
| Bipolar cylindrical coordinates  (u,v,z)\in [0,2\pi )\times (-\infty ,\infty )\times (-\infty ,\infty ) |  {\begin{aligned}x&={\frac {a\sinh v}{\cosh v-\cos u}}\\y&={\frac {a\sin u}{\cosh v-\cos u}}\\z&=z\end{aligned}} |  {\begin{aligned}h_{1}&=h_{2}={\frac {a}{\cosh v-\cos u}}\\h_{3}&=1\end{aligned}} |
| Toroidal coordinates  (u,v,\phi )\in (-\pi ,\pi ]\times [0,\infty )\times [0,2\pi ) |  {\begin{aligned}x&={\frac {a\sinh v\cos \phi }{\cosh v-\cos u}}\\y&={\frac {a\sinh v\sin \phi }{\cosh v-\cos u}}\\z&={\frac {a\sin u}{\cosh v-\cos u}}\end{aligned}} |  {\begin{aligned}h_{1}&=h_{2}={\frac {a}{\cosh v-\cos u}}\\h_{3}&={\frac {a\sinh v}{\cosh v-\cos u}}\end{aligned}} |
| Bispherical coordinates  (u,v,\phi )\in (-\pi ,\pi ]\times [0,\infty )\times [0,2\pi ) |  {\begin{aligned}x&={\frac {a\sin u\cos \phi }{\cosh v-\cos u}}\\y&={\frac {a\sin u\sin \phi }{\cosh v-\cos u}}\\z&={\frac {a\sinh v}{\cosh v-\cos u}}\end{aligned}} |  {\begin{aligned}h_{1}&=h_{2}={\frac {a}{\cosh v-\cos u}}\\h_{3}&={\frac {a\sin u}{\cosh v-\cos u}}\end{aligned}} |
| Conical coordinates  {\begin{aligned}&(\lambda ,\mu ,\nu )\\&\nu ^{2}<b^{2}<\mu ^{2}<a^{2}\\&\lambda \in [0,\infty )\end{aligned}} |  {\begin{aligned}x&={\frac {\lambda \mu \nu }{ab}}\\y&={\frac {\lambda }{a}}{\sqrt {\frac {(\mu ^{2}-a^{2})(\nu ^{2}-a^{2})}{a^{2}-b^{2}}}}\\z&={\frac {\lambda }{b}}{\sqrt {\frac {(\mu ^{2}-b^{2})(\nu ^{2}-b^{2})}{b^{2}-a^{2}}}}\end{aligned}} |  {\begin{aligned}h_{1}&=1\\h_{2}^{2}&={\frac {\lambda ^{2}(\mu ^{2}-\nu ^{2})}{(\mu ^{2}-a^{2})(b^{2}-\mu ^{2})}}\\h_{3}^{2}&={\frac {\lambda ^{2}(\mu ^{2}-\nu ^{2})}{(\nu ^{2}-a^{2})(\nu ^{2}-b^{2})}}\end{aligned}} |

矩形グリッドに施される等角写像。曲がったグリッドの直交性は保持されている。
2次元直交座標の可視化。1つの座標以外を一定にして得られる曲線が、基底ベクトルとともに示されている。基底ベクトルは長さが等しくないことに注意すること：等しい必要はなく、直交していればよい。