How can I calculate for the X coordinate given the Y value of the position? In this example the Y position is -3 units, as seen in the image below.
https://en.wikipedia.org/wiki/Catenary
We know the Span = 100 (width) and the Sag = -10 (height). I'm not sure where to begin in solving for the two points X position marked yellow in the image below.
If for any reason it's easier to solve for the X position on a Parabola rather than a Catenary I'm fine with that, they are similar enough it would be ok.
For a parabola, this isn't hard at all. The parabola equation is then $$ y = -10 + 10 \left (\frac{x}{50}\right)^2 $$
To solve for $y= -3$, you write \begin{align} -3 &= -10 + 10 \left (\frac{x}{50}\right)^2\\ 7 &= 10 \left (\frac{x}{50}\right)^2\\ \frac{7}{10} &= \left (\frac{x}{50}\right)^2\\ \sqrt{\frac{7}{10}} &= \pm \frac{x}{50}\\ 50 \sqrt{\frac{7}{10}} &= \pm x\\ \pm 50 \sqrt{\frac{7}{10}} &= x \end{align} giving $x \approx \pm 41.83.$