Orthogonal Coordinates - Table of Orthogonal Coordinates

Table of Orthogonal Coordinates

Besides the usual cartesian coordinates, several others are tabulated below. 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

\begin{align} x&=r\sin\theta\cos\phi \\ y&=r\sin\theta\sin\phi \\ z&=r\cos\theta \end{align} \begin{align} h_1&=1 \\ h_2&=r \\ h_3&=r\sin\theta \end{align}
Cylindrical polar coordinates

\begin{align} x&=r\cos\phi \\ y&=r\sin\phi \\ z&=z \end{align} \begin{align} h_1&=h_3=1 \\ h_2&=r \end{align}
Parabolic cylindrical coordinates

\begin{align} x&=\frac{1}{2}(u^2-v^2)\\ y&=uv\\ z&=z \end{align} \begin{align} h_1&=h_2=\sqrt{u^2+v^2} \\ h_3&=1 \end{align}
Paraboloidal coordinates

\begin{align} x&=uv\cos\phi\\ y&=uv\sin\phi\\ z&=\frac{1}{2}(u^2-v^2) \end{align} \begin{align} h_1&=h_2=\sqrt{u^2+v^2} \\ h_3&=uv \end{align}
Elliptic cylindrical coordinates

\begin{align} x&=a\cosh u \cos v\\ y&=a\sinh u \sin v\\ z&=z \end{align} \begin{align} h_1&=h_2=a\sqrt{\sinh^2u+\sin^2v} \\ h_3&=1 \end{align}
Prolate spheroidal coordinates

\begin{align} x&=a\sinh\xi\sin\eta\cos\phi\\ y&=a\sinh\xi\sin\eta\sin\phi\\ z&=a\cosh\xi\cos\eta \end{align} \begin{align} h_1&=h_2=a\sqrt{\sinh^2\xi+\sin^2\eta} \\ h_3&=a\sinh\xi\sin\eta \end{align}
Oblate spheroidal coordinates

\begin{align} x&=a\cosh\xi\cos\eta\cos\phi\\ y&=a\cosh\xi\cos\eta\sin\phi\\ z&=a\sinh\xi\sin\eta \end{align} \begin{align} h_1&=h_2=a\sqrt{\sinh^2\xi+\sin^2\eta} \\ h_3&=a\cosh\xi\cos\eta \end{align}
Ellipsoidal coordinates

\begin{align} & (\lambda, \mu, \nu)\\ & \lambda < c^2 < b^2 < a^2,\\ & c^2 < \mu < b^2 < a^2,\\ & c^2 < b^2 < \nu < a^2, \end{align}

where

Bipolar coordinates

\begin{align} x&=\frac{a\sinh v}{\cosh v - \cos u}\\ y&=\frac{a\sin u}{\cosh v - \cos u}\\ z&=z \end{align} \begin{align} h_1&=h_2=\frac{a}{\cosh v - \cos u}\\ h_3&=1 \end{align}
Toroidal coordinates

\begin{align} 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{align} \begin{align} h_1&=h_2=\frac{a}{\cosh v - \cos u}\\ h_3&=\frac{a\sinh v}{\cosh v - \cos u} \end{align}
Conical coordinates

\begin{align} & (\lambda,\mu,\nu)\\ & \nu^2 < b^2 < \mu^2 < a^2 \\ & \lambda \in [0,\infty) \end{align}

 \begin{align} 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)}{a^2-b^2}} \end{align} \begin{align} 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{align}

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