Catenary is Symmetric about Y-Axis

Theorem

Consider a catenary $\CC$.

Let a cartesian plane be arranged so that the $y$-axis passes through the lowest point of the catenary.

$\CC$ exhibits reflectional symmetry in that $y$-axis.

Proof

From Cartesian Equation of Catenary: Formulation $2$, we have the equation of $\CC$:

$y = \dfrac a 2 \paren {e^{x / a} + e^{-x / a} } = a \cosh \dfrac x a$

The result follows directly from Hyperbolic Cosine Function is Even.

$\blacksquare$

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): catenary
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): catenary