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Finding the catenary curve of given arclength that passes through two points requires solving a transcendental equation.

Here's a related question: Is it possible to solve for the parabola that passes through two given points and has given arclength, with a closed-form (algebraic) solution?

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If you want an algebraic solution, there is no clue.

Let's put ourselves in a friendly setting: let the given two points be $(-1,0)$ and $(1,0)$, and let $L$ be the given arc length.

A parabola passing through $(-1,0)$ and $(1,0)$ has equation $$y= a(x^2-1)$$ depending on a positive parameter $a >0 $.

The length of the arc of parabola is

$$\int_{-1}^1 \sqrt{1+ (2ax)^2} dx = \frac{1}{2a}\int_{-2a}^{2a} \sqrt{1+ y^2} dy$$

so you should solve the following equation for $a$

$$2aL= \int_{-2a}^{2a} \sqrt{1+ y^2} dy = \frac{1}{2} \left[ y \sqrt{1+y^2} + \sinh^{-1}y \right]_{-2a}^{2a}$$

which is clearly a trascendental equation.

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