2

I was just now reading the answers to the question "Is a line parallel with itself?" and that lead me to the question are two lines in exactly the same position in fact just one line?

One can think of two lines that happen to be in the same place. Taking two parallel lines, keeping one fixed, moving the other to it, then have it change sides. Are there always two lines? Or do we have an instant when there is only one?

Gonçalo Peres
  • 143
  • If two (straight/curved) lines correspond to the same equation representation, it is useful to treat them as the same object. Analogously, are $y=x+1$ and $2y=2x+2$ the same equation, or merely equivalent representations of the same line? Going off on a tangent: in graph theory, two lines don’t even need to overlap to be considered the same edge (edges are the lines/links that connect two vertices/nodes/points). – ryang Nov 29 '20 at 12:48
  • 1
    I would say: yes and no, whatever answer is most convenient for you right now. But you should shortly note it in your definitions section... – NeitherNor Nov 29 '20 at 20:12

1 Answers1

1

I think so, yes. If two lines are in the exact same position then certainly they have the same set of points. The same line can be parameterised in two different ways, but as As Ryan G pointed out in the comments, the lines represented by the two different parameterisations is in fact the same line.

So in a non-parameterised (usual) context, $y = 2x+1$ and $4x-2y=-2$ are considered to be the "same line" because they represent the same set of points.

With parameterisation: $(x_1(t),y_1(t)): x_1(t) = t$ and $y_1(t) = t $ corresponds to the same set of points as $(x_2(t),y_2(t)): x_2(t) = 2t$ and $y_2(t) = 2t $, despite the value of $(x_1, y_1)$ being different to the value of $(x_2, y_2)$ for a given value of $t$. This means that, despite the two lines having different paths, the two lines are still considered to be the same.

Adam Rubinson
  • 20,052
  • I’d say the two parameterisations are the same line, even if the paths/routes are different (or traced out at different rates). – ryang Nov 29 '20 at 13:14