digplanet beta 1: Athena
Share digplanet:

Agriculture

Applied sciences

Arts

Belief

Chronology

Culture

Education

Environment

Geography

Health

History

Humanities

Language

Law

Life

Mathematics

Nature

People

Politics

Science

Society

Technology

Ellipsoidal coordinates are a three-dimensional orthogonal coordinate system $(\lambda, \mu, \nu)$ that generalizes the two-dimensional elliptic coordinate system. Unlike most three-dimensional orthogonal coordinate systems that feature quadratic coordinate surfaces, the ellipsoidal coordinate system is not produced by rotating or projecting any two-dimensional orthogonal coordinate system.

## Basic formulae

The Cartesian coordinates $(x, y, z)$ can be produced from the ellipsoidal coordinates $( \lambda, \mu, \nu )$ 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 $\lambda$ 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 $\mu$ 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 $\nu$ 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.

## Scale factors and differential operators

For brevity in the equations below, we introduce a function

$S(\sigma) \ \stackrel{\mathrm{def}}{=}\ \left( a^{2} + \sigma \right) \left( b^{2} + \sigma \right) \left( c^{2} + \sigma \right)$

where $\sigma$ can represent any of the three variables $(\lambda, \mu, \nu )$. Using this function, the scale factors can be written

$h_{\lambda} = \frac{1}{2} \sqrt{\frac{\left( \lambda - \mu \right) \left( \lambda - \nu\right)}{S(\lambda)}}$
$h_{\mu} = \frac{1}{2} \sqrt{\frac{\left( \mu - \lambda\right) \left( \mu - \nu\right)}{S(\mu)}}$
$h_{\nu} = \frac{1}{2} \sqrt{\frac{\left( \nu - \lambda\right) \left( \nu - \mu\right)}{S(\nu)}}$

Hence, the infinitesimal volume element equals

$dV = \frac{\left( \lambda - \mu \right) \left( \lambda - \nu \right) \left( \mu - \nu\right)}{8\sqrt{-S(\lambda) S(\mu) S(\nu)}} \ d\lambda d\mu d\nu$

and the Laplacian is defined by

$\nabla^{2} \Phi = \frac{4\sqrt{S(\lambda)}}{\left( \lambda - \mu \right) \left( \lambda - \nu\right)} \frac{\partial}{\partial \lambda} \left[ \sqrt{S(\lambda)} \frac{\partial \Phi}{\partial \lambda} \right] \ + \$
$\frac{4\sqrt{S(\mu)}}{\left( \mu - \lambda \right) \left( \mu - \nu\right)} \frac{\partial}{\partial \mu} \left[ \sqrt{S(\mu)} \frac{\partial \Phi}{\partial \mu} \right] \ + \ \frac{4\sqrt{S(\nu)}}{\left( \nu - \lambda \right) \left( \nu - \mu\right)} \frac{\partial}{\partial \nu} \left[ \sqrt{S(\nu)} \frac{\partial \Phi}{\partial \nu} \right]$

Other differential operators such as $\nabla \cdot \mathbf{F}$ and $\nabla \times \mathbf{F}$ can be expressed in the coordinates $(\lambda, \mu, \nu)$ by substituting the scale factors into the general formulae found in orthogonal coordinates.

• Focaloid (shell given by two coordinate surfaces)

## Bibliography

• Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. p. 663.
• Zwillinger D (1992). Handbook of Integration. Boston, MA: Jones and Bartlett. p. 114. ISBN 0-86720-293-9.
• Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. pp. 101–102. LCCN 6725285.
• Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers. New York: McGraw-Hill. p. 176. LCCN 5914456.
• Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. pp. 178–180. LCCN 5510911.
• Moon PH, Spencer DE (1988). "Ellipsoidal Coordinates (η, θ, λ)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd ed., 3rd print ed.). New York: Springer Verlag. pp. 40–44 (Table 1.10). ISBN 0-387-02732-7.

### Unusual convention

• Landau LD, Lifshitz EM, Pitaevskii LP (1984). Electrodynamics of Continuous Media (Volume 8 of the Course of Theoretical Physics) (2nd ed.). New York: Pergamon Press. pp. 19–29. ISBN 978-0-7506-2634-7. Uses (ξ, η, ζ) coordinates that have the units of distance squared.
 199 videos foundNext >
 Line 22 7b97z57i Datum Coordinate Transformations UFO Ellipsoidal Equations 5g WOW SETIhttp://alienspacesciencenews.wordpress.com/ 7b97z57i 100 videos there are more videos after this one i'll post all then update the #. Math Equation Wow Seti ... Line 22 7b97z59 New Triangulation French Datum Ellipsoid Cartesian Coordinates UFO 5g WOW SETIhttp://alienspacesciencenews.wordpress.com/ 7b97z59 100 videos there are more videos after this one i'll post all then update the #. Math Equation Wow Seti 1... Line 22 7b97z57f Datum Molodensky  Transformations Geographic Coordinates 5g WOW SETIhttp://alienspacesciencenews.wordpress.com/ 7b97z57f 100 videos there are more videos after this one i'll post all then update the #. Math Equation Wow Seti ... Line 22 7b97z60a Converting DD Coordinates Degrees Minutes Seconds UFO Space Travel 5g WOW SETIhttp://alienspacesciencenews.wordpress.com/ 7b97z60a 100 videos there are more videos after this one i'll post all then update the #. Math Equation Wow Seti ... maurieo's Calculating Elements on Map ProjectionsDetermining Projection Coordinates from Ellipsoidal Bearings and Ellipsoidal Distances using New Zealand Projections. Geographic Coordinate System : geoid and spheroid concept : Part - 1My first video......lot of errors......incomplete......bt still wanted to load......improved version will soon be there...... Double Integral Using Polar Coordinates - Part 1 of 3Double Integral Using Polar Coordinates - In this video, I show how to use polar coordinates to set up a double integral to find the volume underneath a plan... calculus 3, 3d-surface sketchingFormer Professor Michael Chamberlain talks about sketching quadratic surfaces (ellipsoids, paraboloids, hyperboloids) in rectangular coordinates in 3-space, ... Line 22 7b97z57h Conversions Geocentric Geographic UFO Mapping 5g WOW SETIhttp://alienspacesciencenews.wordpress.com/ 7b97z57h 100 videos there are more videos after this one i'll post all then update the #. Math Equation Wow Seti ... MicroSurvey CAD - Upgrade Tour 2014 - Coordinate System EngineThe CS-Map coordinate system engine was updated to provide 100% cross-compatibility between MicroSurvey CAD and MicroSurvey FieldGenius. Hundreds of new coor...
 199 videos foundNext >

We're sorry, but there's no news about "Ellipsoidal coordinates" right now.

 Limit to books that you can completely read online Include partial books (book previews) .gsc-branding { display:block; }

Oops, we seem to be having trouble contacting Twitter