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.

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.

The midpoints of the osculating circles of an involute trace out the
original curve (here a circle).

The trace of these midpoints is called evolute. Therefore: The evolute of the involute is the original curve.

The difference of the radii of the osculating circles at inv(t1) and inv(t2) is the arc length of the

original curve c between c(t1) and c(t2). The difference of their midpoints is less, namely

only |c(t1) - c(t2)|. Therefore the smaller circle is inside the larger one.

The trace of these midpoints is called evolute. Therefore: The evolute of the involute is the original curve.

The difference of the radii of the osculating circles at inv(t1) and inv(t2) is the arc length of the

original curve c between c(t1) and c(t2). The difference of their midpoints is less, namely

only |c(t1) - c(t2)|. Therefore the smaller circle is inside the larger one.