Editing Involute (section)

== Examples ==

=== Involutes of a circle ===
[[File:Evolvente-kreis.svg|thumb|Involutes of a circle]]
For a circle with parametric representation <math>(r\cos(t), r\sin(t))</math>, one has
<math>\vec c'(t) = (-r\sin t, r\cos t)</math>.
Hence <math>|\vec c'(t)| = r</math>, and the path length is <math>r(t - a)</math>. 

Evaluating the above given equation of the involute, one gets
:<math>\begin{align}
X(t) &= r(\cos (t+a) + t\sin (t+a))\\
Y(t) &= r(\sin (t+a) - t\cos (t+a))
\end{align}</math>
for the [[parametric equation]] of the involute of the circle.

The <math>a</math> term is optional; it serves to set the start location of the curve on the circle.  The figure shows involutes for <math>a = -0.5</math> (green), <math>a = 0</math> (red), <math>a = 0.5</math> (purple) and <math>a = 1</math> (light blue). The involutes look like [[Archimedean spiral]]s, but they are actually not.

The arc length for <math>a=0</math> and <math>0 \le t \le t_2</math> of the involute is
: <math>L = \frac{r}{2} t_2^2.</math>

[[File:Evolvente-np.svg|300px|thumb|Involutes of a semicubic parabola (blue). Only the red curve is a parabola. Notice how the involutes and tangents make up an orthogonal coordinate system. This is a general fact.]]

=== 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.]]

=== 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.

=== Involutes of a cycloid ===
[[File:Evolvente-zy.svg|250px|thumb|Involutes of a cycloid (blue): Only the red curve is another cycloid]]
The parametric representation <math>\vec c(t) = (t - \sin t, 1 - \cos t)</math> describes a [[cycloid]]. From <math>\vec c'(t) = (1 - \cos t, \sin t)</math>, one gets (after having used some trigonometric formulas)
:<math>|\vec c'(t)| = 2\sin\frac{t}{2},</math> 
and 
:<math>\int_\pi^t 2\sin\frac{w}{2}\,dw = -4\cos\frac{t}{2}.</math> 

Hence the equations of the corresponding involute are
: <math>X(t) = t + \sin t,</math>
: <math>Y(t) = 3 + \cos t,</math>
which describe the shifted red cycloid of the diagram. Hence

* The involutes of the cycloid <math>(t - \sin t, 1 - \cos t)</math> are parallel curves of the cycloid 
: <math>(t + \sin t, 3 + \cos t).</math>
(Parallel curves of a cycloid are not cycloids.)