A catenary is the shape that a hanging flexible chain or cable assumes under its own weight when supported only at its ends. The curve is defined by the hyperbolic cosine function.

Height Formula:

$$ y = a \cdot \cosh\left(\frac{x}{a}\right) $$

Arc Length Formula:

$$ s = a \cdot \sinh\left(\frac{x}{a}\right) $$

Where:

• $y$ is the height of the cable at a horizontal distance $x$ from the lowest point.
• $s$ is the arc length of the cable from the lowest point to $(x, y)$.
• $a$ is the catenary parameter, which determines the "flatness" of the curve. It's equal to $T_0/w$, where $T_0$ is the horizontal tension at the lowest point and $w$ is the weight per unit length.
• $\cosh$ and $\sinh$ are the hyperbolic cosine and sine functions.