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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.## Plotting-function

## Examples

### Involute of a circle

### Involute of a catenary

### Involute of a cycloid

## 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

The evolute of an involute is the original curve less portions of zero or undefined curvature. Compare and

Analytically: if function $r:mathbb\; Rtomathbb\; R^n$ is a natural parametrization of the curve (i.e. $|r^prime(s)|=1$ for all s), then :$tmapsto\; r(t)-tr^prime(t)$ parametrises the involute.

Equations of an involute of a parametrically defined curve are:

$X[x,y]=x-frac\{x\text{'}int\_a^t\; sqrt\; \{\; x\text{'}^2\; +\; y\text{'}^2\; \},\; dt\}\{sqrt\; \{\; x\text{'}^2\; +\; y\text{'}^2\; \}\}$

$Y[x,y]=y-frac\{y\text{'}int\_a^t\; sqrt\; \{\; x\text{'}^2\; +\; y\text{'}^2\; \},\; dt\}\{sqrt\; \{\; x\text{'}^2\; +\; y\text{'}^2\; \}\}$

- In polar coordinates $,\; r,theta$ the involute of a circle has the parametric equation:

- $,\; r=asecalpha$

- $,\; theta\; =\; tanalpha\; -\; alpha$

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.

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 $r(s)=(sinh^\{-1\}(s),cosh(sinh^\{-1\}(s))),$

we have $r^prime(s)=(1,s)/sqrt\{1+s^2\},$

and $r(t)-tr^prime(t)=(sinh^\{-1\}(t)-t/sqrt\{1+t^2\},1/sqrt\{1+t^2\})$.

Substitute $t=sqrt\{1-y^2\}/y$

to get $(\{rm\; sech\}^\{-1\}(y)-sqrt\{1-y^2\},y)$.

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

See Involute gear

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Last updated on Wednesday October 08, 2008 at 09:52:57 PDT (GMT -0700)

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This article is licensed under the GNU Free Documentation License.

Last updated on Wednesday October 08, 2008 at 09:52:57 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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