Hello, I have a "simple" problem for you guys. I am not expert in math and so try to be simple.(adsbygoogle = window.adsbygoogle || []).push({});

I explain the problem by starting with one example. The polar coordinate system has the following main property: with two parameters, rho and theta, each point is described as the intersection of a circle and a straight line. These two curves are orthogonal and so the curvilinear coordinate system is orthogonal.

Another example is if we use ellipses and hyperbolas: they are orthogonal. The transformation is simple:

x = c*cosh(a)* cos(b) (1)

y = c*sinh(a) * sin(b) (2)

where a>0, and 0<=b<=2pi

Now my question. Suppose that we like to use a set of curvilinear coordinate system (plane x-y only) which is orthogonal. Suppose that instead of ellipses or circles one of the type of curves is a closed path defined by (for example) the equation

x^8/A^8 + y^8/B^8 = 1

which is closed curve similar to an ellipse. Ho do I find a similar trasnformation (like equations (1) and (2)) with the property that the curvilinear coordinate system is also orthogonal? Please help me!

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# Closed curve and orthogonal curvilinear coordinate system

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