Elliptic Coordinate System - Basic Definition

Basic Definition

The most common definition of elliptic coordinates is

x = a \ \cosh \mu \ \cos \nu
y = a \ \sinh \mu \ \sin \nu

where is a nonnegative real number and

On the complex plane, an equivalent relationship is

x + iy = a \ \cosh(\mu + i\nu)

These definitions correspond to ellipses and hyperbolae. The trigonometric identity

\frac{x^{2}}{a^{2} \cosh^{2} \mu} + \frac{y^{2}}{a^{2} \sinh^{2} \mu} = \cos^{2} \nu + \sin^{2} \nu = 1

shows that curves of constant form ellipses, whereas the hyperbolic trigonometric identity

\frac{x^{2}}{a^{2} \cos^{2} \nu} - \frac{y^{2}}{a^{2} \sin^{2} \nu} = \cosh^{2} \mu - \sinh^{2} \mu = 1

shows that curves of constant form hyperbolae.

Read more about this topic:  Elliptic Coordinate System

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