In the differential geometry of curves
, an involute
of a smooth curve
is another curve, obtained by attaching an imaginary taut string to the given curve and tracing its free end as it is wound onto that given curve; or in reverse, unwound. It is a roulette
wherein the rolling curve is a straight line containing the generating point.
The evolute of an involute is the original curve less portions of zero or undefined curvature. Compare and
Analytically: if function is a natural parametrization of the curve (i.e. for all s), then :
parametrises the involute.
Equations of an involute of a parametrically defined curve are:
Involute of a circle
is the radius of the circle and
is a parameter
Leonhard Euler proposed to use the involute of the circle for the shape of the teeth of toothwheel gear, a design which is the prevailing one in current use.
Involute of a catenary
The involute of a catenary through its vertex is a tractrix. In cartesian coordinates the curve follows:
Where: t is the angle and sech is the hyperbolic secant (1/cosh(x))
to get .
Involute of a cycloid
One involute of a cycloid is a congruent cycloid. In cartesian coordinates the curve follows:
Where t is the angle and a the radius
The involute of a circle has some properties that makes it extremely important to the gear
industry: If two intermeshed gears have teeth with the profile-shape of involutes (rather than, for example, a "classic" triangular shape), their relative rates of rotation are constant while the teeth are engaged. Also, the gears always make contact along a single steady line of force. With teeth of other shapes, the relative speeds and forces rise and fall as successive teeth engage, resulting in vibration, noise, and excessive wear. For this reason, nearly all modern gear teeth bear the involute shape.
See Involute gear