Parabolic Coordinates - Two-dimensional Parabolic Coordinates

Two-dimensional Parabolic Coordinates

Two-dimensional parabolic coordinates are defined by the equations

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

The curves of constant form confocal parabolae

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

that open upwards (i.e., towards ), whereas the curves of constant form confocal parabolae

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

that open downwards (i.e., towards ). The foci of all these parabolae are located at the origin.

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