I'm creating a program that has two points and a cable hanging between them. I feel like modeling the cable using a catenary would be too hard, so I just simplified it to a parabola.
However, I'm still stuck on making sure that the length of the cable is constant. I don't know how to find a parabola (or any group of parabolas) given its arc length. The main purpose of my program isn't to model the cable, however, so if it's not possible, it's fine.
hint
Let the two points be $$A(-a,0) \text{ and } \; B(a,0)$$ and $ L $ be the length of the cable.
We can take as the equation of the parabola $$y=f(x)=C(x^2-a^2)$$ such that
$$L=\int_{-a}^a\sqrt{1+f'^2(x)}dx$$ $$=2\int_0^a\sqrt{1+4C^2x^2}dx$$
To finish, put $$2Cx=\sinh(t)$$
then
$$C=\frac 1L\int_0^a\cosh^2(t)dt$$
Remark
You can also, take instead a parabola, the more natural curve whose equation is $$y=C(\cosh(x)-\cosh(a))$$