Ellipsoidal Coordinates - Basic Formulae

Basic Formulae

The Cartesian coordinates can be produced from the ellipsoidal coordinates by the equations

x^{2} = \frac{\left( a^{2} + \lambda \right) \left( a^{2} + \mu \right) \left( a^{2} + \nu \right)}{\left( a^{2} - b^{2} \right) \left( a^{2} - c^{2} \right)}
y^{2} = \frac{\left( b^{2} + \lambda \right) \left( b^{2} + \mu \right) \left( b^{2} + \nu \right)}{\left( b^{2} - a^{2} \right) \left( b^{2} - c^{2} \right)}
z^{2} = \frac{\left( c^{2} + \lambda \right) \left( c^{2} + \mu \right) \left( c^{2} + \nu \right)}{\left( c^{2} - b^{2} \right) \left( c^{2} - a^{2} \right)}

where the following limits apply to the coordinates

- \lambda < c^{2} < - \mu < b^{2} < -\nu < a^{2}.


Consequently, surfaces of constant are ellipsoids

\frac{x^{2}}{a^{2} + \lambda} + \frac{y^{2}}{b^{2} + \lambda} + \frac{z^{2}}{c^{2} + \lambda} = 1,

whereas surfaces of constant are hyperboloids of one sheet

\frac{x^{2}}{a^{2} + \mu} + \frac{y^{2}}{b^{2} + \mu} + \frac{z^{2}}{c^{2} + \mu} = 1,

because the last term in the lhs is negative, and surfaces of constant are hyperboloids of two sheets

\frac{x^{2}}{a^{2} + \nu} + \frac{y^{2}}{b^{2} + \nu} + \frac{z^{2}}{c^{2} + \nu} = 1

because the last two terms in the lhs are negative.

Read more about this topic:  Ellipsoidal Coordinates

Famous quotes containing the words basic and/or formulae:

    “There’s one basic rule you should remember about development charts that will save you countless hours of worry.... The fact that a child passes through a particular developmental stage is always more important than the age of that child when he or she does it. In the long run, it really doesn’t matter whether you learn to walk at ten months or fifteen months—as long as you learn how to walk.”
    —Lawrence Kutner (20th century)

    “I don’t believe in providence and fate, as a technologist I am used to reckoning with the formulae of probability.”
    —Max Frisch (1911–1991)