Why is this equation not representing a superellipse$?$

Question

Any equation of the form $$x^n+y^n=1$$ represents a superellipse. But when I plot $$x^{999}+y^{999}=1$$ I get something uncanny.

This is certainly not a superellipse. What is this shape called$?$ Is desmos not working properly as it is not giving the plot of a superellipse$?$ Is this because of the high power in the variables$?$

Any help is greatly appreciated.

Answer 1

In the positive quadrant your picture really does show a quarter of a superellipse (and it nearly looks like a square, as it should, because of the high power).

If $x$ and $y$ are both negative, then obviously $x^{999}+y^{999}$ is negative as well, so it can't equal $1$, and your picture correctly shows no curve in that quadrant.

And if $x$ and $y$ have opposite signs and absolute value greater than $1$, then we must have $y \approx -x$ in order for $x^{999}+y^{999}$ to equal $1$, so you would expect to see something which looks much like the straight line $y=-x$, so your picture is correct there too.

In order to get the full superellipse in all quadrants, you need to use the correct equation, which is $|x|^{999}+|y|^{999}=1$.

Answer 2

Have you tried other odd exponents? For instance, $x^3+y^3=1$ gives a very similar graph, just smooth, without edges, and without a cutoff: the graph goes on toward the upper left and the lower right edges of the screen.

This is a general pattern: superellipses with even exponents start like a circle and get closer to a square as the exponent increases. Superellipses with odd exponents start as a straight line (with $x+y=1$) and develop that hump which starts out smooth but becomes edgier as the exponent increases.

For an exponent of $999$, it should be close to perfectly edgy, which is what you're seeing in your picture. But a computational constraint will now occur: For $|x|$ or $|y|$ larger than $1$, $x^{999}$ and $y^{999}$ grow huge extraordinarily quickly. But desmos has to calculate them in order to compare them to decide wether to draw a point or not. Either the numbers are just too large to compute, or the accuracy of the calculation decreases so far that the comparison isn't accurate anymore.

Disclaimer: I'm not perfectly sure wether the graph is actually called a superellipse, or if the name superellipse is reserved for equations of the form $|x|^n+|y|^n=a.