Abstract
Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which a function in elliptic coordinates can be readily mapped in physical space. Application to Kirchhoff vortex shows that its elliptical core induces two counter-rotating irrotational eddies.
Highlights
Cartesian to Elliptic CoordinatesIn order to invert the functional relation (1), we first eliminate ξ and have x2 cos (η )
Explicit equations are obtained to convert Cartesian coordinates to elliptic coordinates, based on which a function in elliptic coordinates can be readily mapped in physical space
Explicit equations to transform from Cartesian to elliptic coordinates have not been found in the existing literature [5, 6, 7]
Summary
In order to invert the functional relation (1), we first eliminate ξ and have x2 cos (η ). Che Sun: Explicit Equations to Transform from Cartesian to Elliptic Coordinates x2 − y2 = c2 1− p p which becomes c2 p2 + (x2 + y2 − c2) p − y2 = 0. It shows that curves of constant ξ are ellipses. Since q ≤ 0, both roots are real and denoted as (ξ1,ξ2 ) They clearly satisfy e2ξ1 ⋅ e2ξ2 = 1 , which leads to ξ2 = −ξ1 < 0. (4-7) are explicit equations to derive elliptic coordinates from Cartesian grid. They can be realized via computation software such as Matlab
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