AskDefine | Define involute

Dictionary Definition

involute adj
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

User Contributed Dictionary

English

Adjective

  1. difficult to understand, or complicated
  2. In the context of "botany|of a leaf or petal": having the edges rolled inwards
  3. In the context of "biology|of some shells": having a complex pattern of coils

Verb

  1. To roll or curl inwards.

Noun

  1. A curve that cuts all tangents of another curve at right angles; traced by a point on a string that unwinds from a curved object.

See also

Italian

Adjective

involute
  1. Feminine plural form of involuto

Extensive Definition

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

Plotting-function

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:
X[x,y]=x-\fracY[x,y]=y-\frac

Examples

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
With r(s)=(\sinh^(s),\cosh(\sinh^(s)))\,
we have r^\prime(s)=(1,s)/\sqrt\,
and r(t)-tr^\prime(t)=(\sinh^(t)-t/\sqrt,1/\sqrt).
Substitute t=\sqrt/y
to get (^(y)-\sqrt,y).

Involute of a cycloid

One involute of a cycloid is a congruent cycloid. In cartesian coordinates the curve follows:
x=a(t+\sin(t))\, y=a(3+\cos(t))\, Where t is the angle and a the radius

Application

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.

External links

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
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