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The six normal trigonometric functions relate to the unit circle and the coordinates of the endpoints of the radii making some angle $\theta$ with the x-axis (sine represents the y-coordinate, cosine the x-coordinate, and tangent the ratio of the two). Does there exist a analogous set of functions for the unit superellipse with equation $$x^4+y^4=1?$$ I've been struggling to define such functions that are periodic and work for all values of $\theta$, $0$ through $2\pi$.

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$x^2+y^2=1$ can be described as $(\cos{t},\sin{t})$, so $x^{2n}+y^{2n}=1$ can be described as $(\sqrt[n]{\cos{t}},\sqrt[n]{\sin{t}})$ for $0<t<\frac{\pi}{2}$. To generalize this to all $t$, we can multiply by the signum function, which returns $1$ or $-1$ depending on the sign of the input, and we have $$(signum(\cos{t})\sqrt[n]{|\cos{t}|},signum(\sin{t})\sqrt[n]{|\sin{t}|})$$

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  • $\begingroup$ I see that, but how do I make sure that it is defined for all angles from $0$ to $2\pi$? $\endgroup$ – Franklin Pezzuti Dyer May 1 '17 at 23:45
  • $\begingroup$ Well, you could use a piecewise function if nothing else works. $\endgroup$ – Isaac Browne May 1 '17 at 23:46
  • $\begingroup$ That's true, I could define it in that way. $\endgroup$ – Franklin Pezzuti Dyer May 1 '17 at 23:47
  • $\begingroup$ I came up with a slightly nicer method. $\endgroup$ – Isaac Browne May 1 '17 at 23:55
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I would like to expand upon the excellent solution presented by @IsaacBrowne. To that end, I introduce my own generalization of the superellipse, called superconics. The general form is given by

$$f(X) = b(1-c^2|X/a|^q)^{1/p}$$

or its canonical form

$$f(x) = (1-c^2|x|^q)^{1/p}$$

Here, $a$ and $b$ scale the $x$ and $y$ axes, resp. and $c^2=\pm1$. $c^2=1$ corresponds to elliptic and parabolic types and $c^2=-1$ corresponds to hyperbolic types. (More generally, $c^2$ can vary smoothly between.)

The following integral gives the area under the curve, the centroid, moments, and moments of inertia of all the superconics.

$$\int_{-a}^a X^nf^m(X) dX = a^{n+1}b^m\int_{-1}^1 x^nf^m(x) dx$$

First we notice that $a$ and $b$ are superfluous and that we can concentrate on the canonical equation. The results can be scaled appropriately for any $a$ and $b$ afterward according to the factor $a^{n+1}b^m$. Next, and most important, this equation can be solved exactly in terms of known functions, specifically, the Gauss hypergeometric function and the incomplete beta function

$$ \begin{align} \int_{-1}^1 x^nf^m(x) dx & = \frac{2}{n+1} {_2F_1}(-mp,\frac{n+1}{q};1+\frac{n+1}{q};c^2)\\ & = \frac{2}{q} (c^2)^{-\frac{n+1}{q}} B(\frac{n+1}{q},mp+1,c^2) \end{align} $$

Further simplifications accrue when $c^2=1$. To wit, for $m=1, n=0$ we get the area

$$\int_{-1}^1 f(x) dx = \Psi(p,q) = 2\frac{\Gamma(p+1)\Gamma(1+1/q)}{\Gamma(1+1/q)} = \Psi(1/q,1/p)$$

More generally, for arbitrary $m$ and $n$ we substitute $p\to mp$ and $q\to q/(n+1)$. All of the integrals for $c^2=1$ can be expressed solely in terms of the parameter $\Psi$. This is true for bodies of revolution as well as other three-dimensional bodies composed of superconics profiles.

Now, in addition to the above intrinsic equation for the superconics, we can derive the following $parametric$ equation for superconics in the complex plane:

$$z=|\cos^{2/q}(u)|\text{sgn}(\cos(u))+i|(1-c^2\cos(u))^p|\text{sgn}(\sin(u))$$

where $u = [0,\pi]$ for the upper half plane and $u = [0,2\pi]$ for the full plane, e.g., a closed curve.

You can find some illustrations and animations of superconics here Superconics and here A New Twist on Möbius. There is even an animation of a smooth transition from a sphere to a hyperboloid of one and two sheets using only superconics (with variable $c^2$). Here is a small version of that animation. Mind you, we can make similar animations with any superconics shape.

The Hyperspheroid

In the spirit of full disclosure, this is essentially the same response as used here: Polar form of generalized superellipse.

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