Comment by abtinf
8 hours ago
> f(x, y) = 0. With that you can plot the graph of f(x, y) either as a 3D surface with f(x, y) being the height at point (x, y)
If f(x, y) = 0, wouldn’t using f(x, y) for the height just result in a flat graph?
8 hours ago
> f(x, y) = 0. With that you can plot the graph of f(x, y) either as a 3D surface with f(x, y) being the height at point (x, y)
If f(x, y) = 0, wouldn’t using f(x, y) for the height just result in a flat graph?
f(x, y) = 0 is true only for some combinations of x and y. It’s an equation to be solved, not a universal statement like ∀ x, y : f(x, y) = 0, nor a definition like f(x, y) ≔ 0 (or “≝”). The solutions to the equation are the points (x, y) where the graph has height 0. Which points these are depends on how f is defined.
For example, f might be defined as f(x, y) ≔ x² + y² – 1. Then the points (x, y) for which f(x, y) = 0 are those on the unit circle (those for which x² + y² = 1). The graph will have height 0 only for those points.
They're really two different types of equal signs.
f(x,y) = x+y might be better written as f(x,y) := x+y where := means "is defined as". Then f(x,y) = 0 is an equation that expands to x+y = 0, or in familiar intro algebra form, y=-x.
g(x,y) := 0 really is a flat plane.
I'd seen the := in programming for years but always thought it was basically just =. Thank you for your explanation!
I will say that in programming it's commonly used as assignment, which isn't quite the same thing as definition. golang uses it to declare variables so that's pretty close
When we say "f(x, y) = 0" in this context, we also usually have a separate definition for f(x, y) provided, where that f(x, y) is not necessarily 0 at for all x,y. And so this constraint "f(x, y) = 0" means "find pairs of x and y such that it makes f(x, y) become 0".
If "f(x, y) = 0" is actually the definition of f(x, y), then yes, it would be a pretty boring graph.