I have the function
$$f(x,y) = x^{2/3}+y^{2/3} $$
I'm looking to map contour lines.
As usual - I have substituted $f$ with a constant $c$.
Usually what I would do is try to isolate one of the variables to get a better picture, but here it does not seem to work.
A graphing calculator brought up a pinched square shape, but I just can't understand the logical way to get to this shape.
Any ideas how to simplify this problem?
$\qquad\quad$ Geometric shapes described by algebraic equations of the form $|x|^n+|y|^n=r^n$ are called superellipses. For $n=1$, we have a diamond square, determined by four straight line segments. For $n>1$, these $4$ lines begin to bend outwards, forming a convex shape. Thus, $n=2$ yields a circle, for instance. Letting $n<1$ makes the lines bend inwards, and thus the resulting shape is concave. The case $n=\dfrac23$ in particular is called an astroid. Note that the irreducible even numerator makes the absolute value signs redundant.