Catenary Curves

• The curve that a haning flexible wire or chain forms when it is supported at its ends.
• Galileo had originally claimed that the curve of the chin would be a parabola.

Formula for a catenary:

x(t) = t

y(t) = 1/2 a [e^(t/a) + e^(-t/a)] = a cosh (t/a)

t = 0 corresponds to the vertex

• Cesaro equation expresses a curve in terms of arc length "s", and curvature "k".

(s^2 + a^2)k = -a

s(t) = a sinh (t/a)

k(t) = -1/a sech^2 (t/a)

o(t) = -2 tan^-1 [tanh (t/2a)]

• The slope is proportional to the arc length that is measured from the center of symmetry.
• This freely hanging curve has the least potential gravitational energy. (The potential energy of a mass "m" is [mgh])
• "cosh" is a hyperbolic cosine function.
• These curves arise because they are the solution of several problems from the calculus of variations.
• They share some properties with corresponding circular functions.
• They arise in many problems in mathematics in which there are integrals such as (1 +x^2)-1/2 and circular functions are (1 - x^2)-1/2.