Mathieu functions toolbox for Scilab


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Elliptic and Cartesian coordinates

Elliptic and Cartesian coordinates

Almost all tasks, involving Mathieu functions have elliptical geometry.

Let us assume an ellipse in the (x, y) plane whose equation is (1)


where a > b are major and minor semi-axes of ellipse. The semi-focal distance f is given by (2)


Conversion between Cartesian and elliptic coordinates are given by equation (3) [1]


from which one can obtain the forward and backward conversion formulas:

Conversion from elliptic to Cartesian coordinates Conversion from Cartesian to elliptic coordinates
(function mathieu_ell2cart) (function mathieu_cart2ell)

For better understanding of elliptical coordinates you can use the demo named "Elliptic and Cartesian coordinates" we have an ellipse with semi-axes a = 5, b = 4 and semi-focal f = 3.

This demo uses mathieu_ell2cart function.

You can launch it from Scilab menu "? -> Scilab Demonstrations -> Mathieu functions -> Elliptic and Cartesian coordinates".


  1. N. W. McLachlan, Theory and Application of Mathieu Functions, Oxford Univ. Press, 1947.