image of Elliptic cylindrical

Elliptic cylindrical

3D Coordinate Systems

Description

Elliptic cylindrical coordinates are a three-dimensional orthogonal coordinate system that results from projecting the two-dimensional elliptic coordinate system in the perpendicular z-direction. Hence, the coordinate surfaces are prisms of confocal ellipses (see Ellipse) and hyperbolae (see Hyperbola).

Object definitions

Mapping

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

Constants

Constants of Elliptic cylindrical
\{a\}
<math> <mrow> <mo>{</mo> <mi>a</mi> <mo>}</mo> </mrow> </math>
{a}
[a]

Cite this as:

3D Coordinate Systems: Elliptic cylindrical from Differential Geometry Library. http://digi-area.com/DifferentialGeometryLibrary/3DCoordinateSystems/Elliptic-Cylindrical.php

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