Radius of Curvature of Catenary

a

x

show evolute

Consider the catenary

y=acosh(x/a)

(blue curve). Let

C

be an arbitrary point on the catenary and let

B

be the point where the normal to the catenary meets the

x

axis. Then the radius of curvature of the catenary at

C

is equal to the distance from

C

to

B

, that is,

|PC|=|CB|

, where

P

is the center of the osculating circle to the catenary at

C

.