Involute

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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.

The evolute of an involute is the original curve less portions of zero or undefined curvature. Compare Media:Evolute2.gif and Media:Involute.gif

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 : $t\mapsto r(t)-tr^\prime(t)$ parametrizes the involute.

1 Parametric curve
2 Examples
3 Application
4 External links

Parametric curve

Equations of an involute of a parametrically defined curve are:

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

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

Examples

The involute of a circle
(in reverse, by unwinding)

The involute of a catenary, a tractrix.

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

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

Involute of a cycloid

One involute of a cycloid is a congruent cycloid. In cartesian coordinates the curve follows:

$x=r(t-sin(t))\,$

$y=r(1-cos(t))\,$

Where t is the angle and r is 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