Circle And Involutes

If one unreavels a thread from a wheel, then the endpoint of the thread moves on the involute of a circle.

Involutes of a parametrized curve c(t) are computed
as follows: Pick a starting value t0 and for t > t0
draw the backwards tangent of length (t-t0):
    inv(t) = c(t) - (t-t0)*c'(t).

The shape of the teeth of gear wheels for heavy
machinery is given by circle involutes. The main
advantage of such gear wheels is that the ratio of the
angular velocities of the driving and the driven wheel
is constant. This avoids destructive vibrations.