Editing Involute (section)

=== Involutes of a semicubic parabola ===
The [[parametric equation]] <math>\vec c(t) = (\tfrac{t^3}{3}, \tfrac{t^2}{2})</math> describes a [[semicubical parabola]]. From <math>\vec c'(t) = (t^2, t)</math> one gets <math>|\vec c'(t)| = t\sqrt{t^2 + 1}</math> and <math>\int_0^t w\sqrt{w^2 + 1}\,dw = \frac{1}{3}\sqrt{t^2 + 1}^3 - \frac13</math>. Extending the string by  <math>l_0={1\over3}</math> extensively simplifies further calculation, and one gets
: <math>\begin{align} X(t)&= -\frac{t}{3}\\
Y(t) &= \frac{t^2}{6} - \frac{1}{3}.\end{align}</math>

Eliminating  {{mvar|t}} yields <math>Y = \frac{3}{2}X^2 - \frac{1}{3},</math> showing that this involute is a [[parabola]]. 

The other involutes are thus [[parallel curves]] of a parabola, and are not parabolas, as they are curves of degree six (See {{slink|Parallel curve|Further examples}}).
[[File:Involute.gif|thumb|The red involute of a catenary (blue) is a tractrix.]]