I would like to plot this function:
$\sqrt[\gamma]{(\frac{x}{a})^{\gamma}+(\frac{y}{b})^{\gamma}}=\mu \,$ where $\, \gamma, a,b\in \mathbb{R} \,$ and $\, \mu \in \mathbb{Z}$
I have tried to plot this function in MATLAB considering $\gamma=0.43$, $ a=0.2$, $b=0.55$, $\mu=2$, but it looks so ugly. I am wonder if my method is correct. Here is what I do:
First) Rearrange the equation .
Second) Plot it with a simple code:
$y=\pm b\times\sqrt[{\gamma}]{{\mu^{\gamma}-(\frac{x}{a})^{\gamma}}}$
Here is a code I use in MATLAB to plot this function:
g=0.2; a =0.4; b=0.6; m=1;
x= - a*m :0.01: a*m;
y= +b*(m^g-(x/a).^g).^(1/g);
yy= -b*(m^g-(x/a).^g).^(1/g);
plot(x,y,x,yy)
The result should be a nice closed shape called superellipse. But it is not.
Is what I've done mathematically correct?
Firstly your function needs absolute values in it if you want to evaluate the fractional powers correctly.
$$\sqrt[\gamma]{(\frac{|x|}{a})^{\gamma}+(\frac{|y|}{b})^{\gamma}}=\mu \,$$
Also your function does not need the $\mu$ term as it can be rearranged as:
$$\sqrt[\gamma]{(\frac{|x|}{a\cdot\mu})^{\gamma}+(\frac{|y|}{b\cdot\mu})^{\gamma}}=1$$
Regardless your function can then be written as:
$$y=\pm b\cdot\mu\times\sqrt[{\gamma}]{{1-(\frac{|x|}{a\cdot\mu})^{\gamma}}}$$
This shows that your range of $x$ should be $[-a\cdot\mu,a\cdot\mu]$ whereas in your example you have used a much bigger range for $x$. This may be the cause of your problem.
Lastly most computer programs will have difficulty near $x=0$ as the slope of the curve there is approaching infinity and any small increment in the $x$ direction can result in a large change in the $y$ which may be calculated poorly. This is probably also contributing to your problem.
Without a picture to tell exactly what you graph looks like there isn't much more I can say.