# 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

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

### Noun

### See also

## Italian

### Adjective

involute- 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

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.See Involute
gear

## 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