# 拋物線座標系

（重定向自拋物線坐標系

## 二維拋物線坐標系

$x = \pm\,\sigma \tau$
$y = \frac{1}{2} \left( \tau^{2} - \sigma^{2} \right)$

$\sigma=\sqrt{ - y +\sqrt{x^2+y^2}}$
$\tau=\sqrt{y +\sqrt{x^2+y^2}}$

$2y = \frac{x^{2}}{\sigma^{2}} - \sigma^{2}$

$2y = - \frac{x^{2}}{\tau^{2}} + \tau^{2}$

## 二維標度因子

$h_{\sigma} = h_{\tau} = \sqrt{\sigma^{2} + \tau^{2}}$

$dA = \left( \sigma^{2} + \tau^{2} \right) d\sigma d\tau$
$\nabla^{2} \Phi = \frac{1}{\sigma^{2} + \tau^{2}} \left( \frac{\partial^{2} \Phi}{\partial \sigma^{2}} + \frac{\partial^{2} \Phi}{\partial \tau^{2}} \right)$

## 三維拋物線坐標系

$x=\sigma\tau\cos\phi$
$y=\sigma\tau\sin\phi$
$z=\frac{1}{2}\left(\tau^{2} - \sigma^{2}\right)$

$\tan \phi = \frac{y}{x},\qquad 0\le \phi\le 2\pi$

$\sigma=\sqrt{ - z +\sqrt{x^2+y^2+z^2}}$
$\tau=\sqrt{z +\sqrt{x^2+y^2+z^2}}$
$\phi =\tan^{ - 1}\frac{y}{x}$

$2z = \frac{x^{2} + y^{2}}{\sigma^{2}} - \sigma^{2}$

$2z = - \frac{x^{2} + y^{2}}{\tau^{2}} + \tau^{2}$

## 三維標度因子

$h_{\sigma} = \sqrt{\sigma^2+\tau^2}$
$h_{\tau} = \sqrt{\sigma^2+\tau^2}$
$h_{\phi} = \sigma\tau\,$

$dV = h_\sigma h_\tau h_\phi = \sigma\tau \left( \sigma^{2} + \tau^{2} \right)\,d\sigma\,d\tau\,d\phi$
$\nabla^2 \Phi = \frac{1}{\sigma^{2} + \tau^{2}}\left[ \frac{1}{\sigma} \frac{\partial}{\partial \sigma} \left( \sigma \frac{\partial \Phi}{\partial \sigma} \right) + \frac{1}{\tau} \frac{\partial}{\partial \tau} \left( \tau \frac{\partial \Phi}{\partial \tau} \right)\right] + \frac{1}{\sigma^2\tau^2}\frac{\partial^2 \Phi}{\partial \phi^2}$

## 第二種表述

$\eta ={ - z+\sqrt{x^2+y^2+z^2}}$
$\xi ={z+\sqrt{x^2+y^2+z^2}}$
$\phi =\arctan {y \over x}$

$\eta = - z + \sqrt{ x^2 + z^2}$
$\xi =z + \sqrt{ x^2 + z^2}$

$\left. z \right|_{\eta = c} = {x^2 \over 2 c} - {c \over 2}$

$\left. z \right|_{\xi = b} = {b \over 2} - {x^2 \over 2 b}$

${x^2 \over 2 c} - {c \over 2} = {b \over 2} - {x^2 \over 2 b}$

$x=\sqrt{bc}$

$z_c= {bc\over 2 c} - {c \over 2} = {b - c \over 2}$

$\frac{dz_c}{dx}=\frac{x}{c}=\sqrt{\frac{b}{c}}=s_c$

${dz_b\over dx}= - {x \over b}= - \sqrt{c\over b}= s_b$

$s_c s_b= - 1$

$x = \sqrt{\eta\xi}\ \cos \phi$
$y = \sqrt{\eta\xi}\ \sin \phi$
$z =\frac{1}{2}(\xi - \eta)$

## 參考文獻

• Margenau H, Murphy GM. The Mathematics of Physics and Chemistry. New York: D. van Nostrand. 1956: pp. 185–186.
• Korn GA, Korn TM. Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. 1961: p. 180.
• Sauer R, Szabó I. Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. 1967: p. 96.
• Zwillinger D. Handbook of Integration. Boston, MA: Jones and Bartlett. 1992: p. 114. ISBN 0-86720-293-9.
• Moon P, Spencer DE. Parabolic Coordinates (μ, ν, ψ). Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions corrected 2nd ed., 3rd print ed. New York: Springer-Verlag. 1988: pp. 34–36 (Table 1.08). ISBN 978-0387184302.
1. ^ Menzel, Donald H. Mathematical Physics. United States of America: Dover Publications. 1961: pp. 139. ISBN 978-0486600567 （English）.