Comment by rustybolt
4 hours ago
Ouch, this hurts to read. It's not novel and lacks a very basic understanding of math.
The graph of y/(x^2+y^2)=(x+1)/(x^2+y^2) by definition contains the points that satisfy this equation. This is exactly the set of points for which y = x + 1.
The "fuzzy" graph is just coloring the difference between the left hand side and right hand side. This is very basic, not new, and it's definitely not "the graph of y/(x^2+y^2)=(x+1)/(x^2+y^2)".
It confused me a lot - I cancelled the denominators in my head too.
But then I realised they're just plotting
y/(x^2+y^2) - (x+1)/(x^2+y^2) = c
and colouring by c (i.e. a heatmap, as others have mentioned in the thread).
That's why you get a more interesting image than you'd get with y - (x + 1) = c
Why would you say it's not a graph of y/(x^2+y^2)=(x+1)/(x^2+y^2)? I would argue that a conventional/binary graph is also not a "pure" representation of the equation, but rather one possible representation - one that runs it through a "left_side == right_side?" boolean filter. In fact, there is no way to visualize an equation with doing something to it.
There's an equal sign in the equation. That means it is true when y = x + 1. There's no filter we're applying, that's literally what the equation says. What you plot is f(x,y) = (y-x-1)/(x^2+y^2). The line plot is when that equals zero, the fuzzinss of it is when it doesnt. But notice that f(x,y)=0 is exactly equivalent to y=x+1. They're exactly the same. Thus, when you're plotting the fuzzy graph it is definitively _not_ a plot of y=x+1, it's a plot of z=(y-x-1)/(x^2+y^2) and those are not the same thing.
We'd only need to "apply a filter" to get the line graph if we started with z(x,y), but that's not what you wrote
I think the parent basically sensed that you're not a trained mathematician and is trying to throw their middle-school math textbook at you.
The simplest definition of a "graph of a function" is that it's a representation of the points satisfying some underlying equality. Your plot isn't that. A more conventional name would be a heatmap: a plot of a function that takes two parameters - x and y coordinates - and then assigns a third value (color) to each.
I don't think the distinction is all that interesting. They're both function plots.
The "graph" of a function is formally defined as exactly those points that make the equation true. https://en.wikipedia.org/wiki/Graph_of_a_function Granted, the graph is only one visualization of a function, and not the only valuable one.
Of course we also have to remember that functions are not the same as equations, and a given function, or more generally relation, can be represented by multiple different equations. For a trivial example, multiply both sides of your slashdot equation by a constant, or add x*y.