10182

# Constant Coordinate Curves for Elliptic Coordinates

This Demonstration shows curves of constant coordinate values for the elliptic coordinate system in two dimensions. These curves are semi-hyperbolas and ellipses, the latter having foci at . As you drag the locator in the plane, the curves are redrawn so they pass through that point. Holding the mouse over the curve shows which variable is constant along that curve, and holding it over the point gives the actual values of the variables. You can vary the interfocal separation, ; at the elliptic coordinates are equivalent to polar coordinates.

### DETAILS

Two-dimensional elliptic coordinates may be defined by , , for and . The curves of constant and are ellipses and hyperbolas, respectively.
The inverse relation can be expressed , , where is the distance from the left/right focus. The slightly ungainly factor is necessary here to ensure that the correct (upper/lower) half-plane is chosen.
In the limit , the elliptic coordinates reduce to polar coordinates . The correspondence is given by and (note that itself becomes infinite as ).
Three-dimensional generalizations of the elliptic coordinates are the oblate and prolate spheroidal coordinates, elliptic cylindrical coordinates, and ellipsoidal coordinates.

### PERMANENT CITATION

 Share: Embed Interactive Demonstration New! Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details » Download Demonstration as CDF » Download Author Code »(preview ») Files require Wolfram CDF Player or Mathematica.

#### Related Topics

 RELATED RESOURCES
 The #1 tool for creating Demonstrations and anything technical. Explore anything with the first computational knowledge engine. The web's most extensive mathematics resource. An app for every course—right in the palm of your hand. Read our views on math,science, and technology. The format that makes Demonstrations (and any information) easy to share and interact with. Programs & resources for educators, schools & students. Join the initiative for modernizing math education. Walk through homework problems one step at a time, with hints to help along the way. Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet. Knowledge-based programming for everyone.