what is a double point in a curve? Is it located only at origin?

Question

I want to know whether a double point like a node or a cusp is only located at origin.

Answer 1

Say you have a curve given by $f(x, y) = 0$ (like $x^3-y^2 = 0$, or $y^2-x^2(x+1) = 0$), with a singular point (cusp or double point) at the origin. Then the curve given by $g(x, y) = f(x-1, y-1)$ will be a translation by one unit vertically and one unit horizontally, moving the singular point to $(1,1)$.

One could even say that this isn't a result of moving the curve at all, but rather of moving the coordinate axes, illustrating how arbitrary the exact coordinates of points (including the origin) often can be.

You have mostly seen examples of curves with singular points is because those usually have the simplest formulas to write down and read, so they're the easiest examples to make and understand.