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Limits of Desmos, Part 2

A few years ago, I posted about some limitations of the Desmos online graphing software. Last week, I was playing with Desmos again and found a couple more.

Scaling a Line

Starting from any true equation, manipulating both sides in the same way yields another true equation. Often, we use this idea to isolate variables when solving an equation.

Suppose we wanted to graph the equation $2x+y=3$. The graph is the set of all pairs $(x,y)$ that make the equation true.

If the equation is true for a certain $x$ and $y$, then multiplying both sides by the same amount does not change the fact; we expect the same graph. In Desmos, I graphed $A\cdot (2x+y)=A\cdot 3$ and put a slider on $A$ so that I could drag its value around. For the most part, I got what I expected. The case with $A=3.2$ is depicted below. Notably, the line is the same.

However, there was one $A$-value that didn’t work out that way: $A=0$. Desmos produced an empty graph.

But the graph should not have been empty. When $A=0$, I’d multiplied both sides of my equation by $0$, so it simplified to $0=0$. The graph of that new equation is the set of pairs $(x,y)$ that make it true. But no matter what the $x$- and $y$-values are, $0$ is equal to $0$. That means that every point should be on the graph, just like the graph of $x^2>-1$ below.

Squircles

Last week’s post on the Cre8Math blog mentioned squircles, a family of curves representing a transition from a circle to a square. In algebra, we learn that the equation $x^2+y^2=1$ describes a circle. Less commonly-known is the fact that $x^2+y^2-x^2y^2=1$ describes a square (with some extra bits that we ignore). To see why, we can move some terms around and factor:

\begin{align*} 0 &= x^2y^2-x^2-y^2+1 \\ 0 &= (x^2-1)(y^2-1) \end{align*}

For the product of $x^2-1$ and $y^2-1$ to be $0$, one of those factors has to be $0$. Since $x^2-1=0$ only when $x=\pm 1$ and similarly for $y$, the graph of this equation consists of two vertical and two horizontal lines. The central square where these lines cross is what we are interested in.

A squircle is somewhere between a square and a circle. Squircles have the equation $x^2+y^2-s\cdot x^2y^2=1$ for some value of $s$ between $0$ and $1$ (observe that $s=0$ is the circle and $s=1$ gives the square). We can get a good sense of how squircles work by typing this equation into Desmos and putting $s$ on a slider. Here is the circle with $s=0$.

And here is a squircle with $s\approx 0.3$. Notice that there are some extra pieces of the graph in the corners, but we only care about the central shape.

But when we get to $s=1$, we don’t get the square we expect: the vertical lines are missing!

If you want to see the whole $s=1$ squircle, it looks like you need to separately graph $x^2-1=0$ and $y^2-1=0$, make them the same color, and pretend it’s just one graph.

Conclusion

I point these bugs out not to criticize Desmos. It’s a fantastic piece of software that I use extensively in my classroom. But they serve as a reminder that even with the best pieces of technology, we still need to think critically about what we are seeing and whether it is giving the correct picture.

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