Superellipse and a good coordinate system

by traianus
Tags: coordinate, superellipse
 P: 80 Hello guys, at the link http://mathworld.wolfram.com/Superellipse.html you can find the definition of superellipse. Now consider the particular super ellipse $$\frac{x^{2n}}{A^{2n}} +\frac{y^{2n}}{B^{2n}} = 1$$ In which A,B, are constant and n is a positive integer. What is the coordinate system that has two families of curves in which one represents the superellipse and the other one is perpendicular to it? In the particular case of A = B, n = 1, the coordinate system is $$x = r\cos{\varphi}$$ $$y = r\sin{\varphi}$$ If A is different than B but still n =1, the coordinate system is similar but it involves also hyperbolic sine and cosine and one family of curves is the generic ellipse and the other family is the generic hyperbola and they are perpendicular to each other, like it happens in the polar coordinates. So, is there a similar curvilinear coordinate system for the particular superellipse I described?
 P: 38 What about: $$x = A (\cos \varphi)^{\frac{1}{n}}$$ $$y = B (\sin \varphi)^{\frac{1}{n}}$$ ?
 P: 80 This is a parametric representation of the superellipse, not a coordinate system. In fact, you have only a "free" parameter $$\varphi$$ and you must have two parameters, like for polar coordinates, where you have $$\varphi$$ and $$r$$.
 P: 38 Superellipse and a good coordinate system Ok...so consider then: $$x = A\, r [\cos \varphi]^{\frac{1}{n}}$$ $$y = B\, r [\sin \varphi]^{\frac{1}{n}}$$ from which we then get: $$\frac{x^{2n}}{A^{2n}} +\frac{y^{2n}}{B^{2n}} = r^{2n}$$ Let $\{\mathbf{e}_1, \mathbf{e}_2} \}$ form an orthonormal basis for the rectangular coordinate system and let $\mathbf{w} = x\, \mathbf{e}_1 + y\, \mathbf{e}_2\;$. So then, $$\mathbf{w} = r (A [\cos \varphi]^{\frac{1}{n}} \mathbf{e}_1 + B [\sin \varphi]^{\frac{1}{n}} \mathbf{e}_2)$$ To find a tangent vector to some superellipse given by fixed $r\,$, we find: $$\frac{\partial \mathbf{w}}{\partial \varphi} = \frac{r}{\sin \varphi \cos \varphi} (-\frac{A\, \sin^2 \varphi}{n} [\cos \varphi]^{\frac{1}{n}} \mathbf{e}_1 + \frac{B\, \cos^2 \varphi}{n} [\sin \varphi]^{\frac{1}{n}} \mathbf{e}_2)$$ And I have no idea where I'm going with this but I'm having fun with the TeX stuff :p