# Elliptic

## 2D Coordinate Systems

#### Description

The Elliptic coordinate system is a two-dimensional orthogonal coordinate system in which the coordinate lines are confocal ellipses and hyperbolae (see Hyperbola).
Elliptic coordinates form the basis for several sets of three-dimensional orthogonal coordinates (see Elliptic cylindrical, Prolate spheroidal, Oblate spheroidal).

### Mapping

\{x\to \cosh (u) \cos (v),y\to \sinh (u) \sin (v)\}
$<mrow> <mo>{</mo> <mrow> <mrow> <mi>x</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mrow> <mi>cosh</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <mo>&#8290;</mo> <mrow> <mi>cos</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mrow> <mo>,</mo> <mrow> <mi>y</mi> <semantics> <mo>&#8594;</mo> <annotation encoding='Mathematica'>&quot;\[Rule]&quot;</annotation> </semantics> <mrow> <mrow> <mi>sinh</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>u</mi> <mo>)</mo> </mrow> <mo>&#8290;</mo> <mrow> <mi>sin</mi> <mo>&#8289;</mo> <mo>(</mo> <mi>v</mi> <mo>)</mo> </mrow> </mrow> </mrow> </mrow> <mo>}</mo> </mrow>$
{x -> Cos[v]*Cosh[u], y -> Sin[v]*Sinh[u]}
[x = cos(v)*cosh(u), y = sin(v)*sinh(u)]

### Cite this as:

2D Coordinate Systems: Elliptic from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/2DCoordinateSystems/Elliptic.php