1 especially of petals or leaves in bud; having margins rolled inward [syn: rolled]
2 (of some shells) closely coiled so that the axis is obscured
- Feminine plural form of involuto
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 Media:Evolute2.gif and Media:Involute.gif
Analytically: if function r:\mathbb R\to\mathbb R^n is a natural parametrization of the curve (i.e. |r^\prime(s)|=1 for all s), then :t\mapsto r(t)-tr^\prime(t) parametrises the involute.
Equations of an involute of a parametrically defined curve are:
Involute of a circle
- In polar coordinates \, r,\theta the involute of a circle has the parametric equation:
\, r=a\sec\alpha \, \theta = \tan\alpha - \alpha where \, a is the radius of the circle and \, \alpha 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:
x=t-\tanh(t)\, y=\rm sech(t)\, Where: t is the angle and sech is the hyperbolic secant (1/cosh(x)) Derivative
we have r^\prime(s)=(1,s)/\sqrt\,
to get (^(y)-\sqrt,y).
Involute of a cycloid
x=a(t+\sin(t))\, y=a(3+\cos(t))\, Where t is the angle and a the radius
ApplicationThe 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
involute in Bosnian: Evolventa
involute in German: Evolvente
involute in Estonian: Evolvent
involute in French: Involute
involute in Hungarian: Evolvens
involute in Dutch: Evolvente
involute in Japanese: インボリュート曲線
involute in Polish: Ewolwenta
involute in Russian: Эвольвента
involute in Slovak: Evolventa
involute in Swedish: Cirkelevolvent
involute in Chinese: 漸伸線