Given that the Cartesian equation for a Superellipse is: $$ \left | \frac{x}{a} \right |^{m} + \left | \frac{y}{b} \right |^{n} = 1 $$ How do we reach the (Widely accepted) parametric form that is: $$ x=a\cdot \cos^{2/m}t $$ $$ y=b\cdot \sin^{2/n}t $$
I believe you would start by taking the Cartesian equation and substitute x and y with: $a\cdot \cos t$ and $b\cdot \sin t$ respectively, which would give: $$ \left | \cos t \right |^{m} + \left | \sin t \right |^{n} = 1 $$
Going from here I suspect you would use the Pythagorean identity of: $$ \cos^{2}t + \sin^{2}t = 1 $$
...To reach the final equation, but I don't really follow how. Specifically, where does the number 2 in the exponent of 2/m and 2/n come from? And how are the variables a and b "reintroduced" at the end?
If I plug in the Cartesian equation into something like Geogebra it plots the superellipse from the get go, but to reach the same results through programming it feels like I'm just introducing these exponents to produce the desired result, and I would love to understand how we come to these seemingly magic numbers. (I also notice that I can get similar results with a different quotient than 2 in the exponent, this seems to control the sharpness of the curve...)
I realize this is probably rudimentary, but I've been struggling to find any resources that explain this step in more detail, so thanks for your help.
To go from the cartesian to the parametric equations, write the left hand side as $\cos^2 t + \sin^2 t$ to get $$\left|\dfrac {x}{a}\right |^{m} + \left|\dfrac {y}{b}\right |^{n} = \cos^2 t + \sin^2 t$$
Then, let $\left|\dfrac {x}{a}\right |^{m} = \cos^2 t$ and take the $\dfrac {1}{m}$th root to get $$\frac {x}{a} = (\cos t)^{\frac {2}{m}}$$ Multiply by $a$ and you get $$x = a \cdot \cos ^{\frac {2}{m}} t$$ Do the same operation for $\left|\dfrac {y}{b}\right |^{n} = \sin^2 t$ and take the $\dfrac {1}{n}$th root to get $$\frac {y}{b} = (\sin t)^{\frac {2}{n}}$$ and then multiply by $b$ to get $$y = b \cdot \sin ^{\frac {2}{n}} t$$ Thus, you have your parametric equations $$x = a \cdot \cos ^{\frac {2}{m}} t \\ y = b \cdot \sin ^{\frac {2}{n}} t$$
Check (doing the reverse):
In the parametric equation, if you divide each by $a$ and $b$ respectively, you will have $$\frac {x}{a} = (\cos t)^{\frac {2}{m}} \\ \frac {y}{b} = (\sin t)^{\frac {2}{n}}$$ All you need to do is raise each side to the $\dfrac {m}{2}$ and $\dfrac {n}{2}$ powers to get $$\left|\dfrac {x}{a}\right |^{\frac {m}{2}} = \cos t \\ \left|\dfrac {y}{b}\right|^\frac {n}{2} = \sin t$$
Then you can use the Pythagorean theorem to get the original equation.
Addendum: According to Wikipedia, the equation for the superellipse is $$\left|\dfrac {x}{a}\right |^{n} + \left|\dfrac {y}{b}\right |^{n} = 1$$ but it does turn out some nice shapes with the equation you have with $m \gt n$ or $n \gt m$.