I'm trying to compute the chain/rope/string curve between two points and a given length. I followed the instructions as answered here but I have a last step to accomplish. The equations I'm solving assume the catenary is right on the y-axis. How do I find where in the curve my initial two points were? I want to plug in my x-coordinates and get the corresponding y-coordinates.
Horizontal shifts are accomplished by making a change of variable $u = x - \sigma_r$ where your former variable was $u$, the amount you want to shift right by is $\sigma_r$, and $x$ is the variable as you want it displayed in the future.
For example., the line $y = 2u - 1$ can be shifted 3 units to the right by doing $y = 2(x-3) - 1 = 2x - 7$.
To double check: $(1,1)$ was a point in the original, and $(4,1)$ is a point in the new.
The same method works to horizontal shift any curve, be it a line, a quadratic, a catenary, or whatever.
To shift vertically, it works similarly, just replace your former codomain variable with $y-\sigma_u$ where $\sigma_u$ is how much you shifted up.
tldr: its better not to think about moving the lines themselves as it is to move the axis via a change of variable.