Editing Involute (section)

=== Involutes of a catenary ===
For the [[catenary]] <math>(t, \cosh t)</math>, the tangent vector is <math>\vec c'(t) = (1, \sinh t)</math>, and, as <math> 1 + \sinh^2 t =\cosh^2 t,</math> its length is <math>|\vec c'(t)| = \cosh t</math>. Thus the arc length from the point {{math|(0, 1)}} is  <math>\textstyle\int_0^t \cosh w\,dw = \sinh t.</math>

Hence the involute starting from {{math|(0, 1)}} is parametrized by
: <math>(t - \tanh t, 1/\cosh t),</math>
and is thus a [[tractrix]].

The other involutes are not tractrices, as they are parallel curves of a tractrix.