Catenary is Symmetric about Y-Axis
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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