Editing Involute (section)

=== 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.]]