Investigating the geometric patterns of $x^k+y^k=r^k$ for $k \in N$

Question

I was investigating the graphs of equations of the form $x^k+y^k=r^k$. I am not sure how to ask this so I will try to simplify the problem first.

For simplicity sake lets let $r=2$, now

For $k=1$, I get a line.

$x+y=2$.

For $k=2$, I get a circle with radius 2.

$x^2 + y^2= 4$.

Is there a name for the image I get when k=3?

$x^3+y^3=8$?

Are there more names of the geometric figures for higher powers of $k \in N$?

I have included a picture of these two cases where $k=3$ and $k=4$

Answer 1

For even $k$, these are Lamé Curves also known as hyperellipses.

For odd $k$, you have part of a Lamé Curve in quadrant 1 and the curve asymptotically approaches the line $y = -x$ from above in quadrants two and four. I do not know of a name for the full curves for odd $k$. (They are examples of superelliptic curves, but that is almost useless as a label.)