Involute

From Wikipedia, the free encyclopedia

In the differential geometry of curves, an involute of a smooth curve is another curve, obtained by attaching a string to the curve and tracing the end of the string as it is wound onto the curve. It is a roulette wherein the rolling curve is a straight line containing the generating point.

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.

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

Equations of an involute of a parametricaly defined curve are:

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

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

1 Examples
2 Application
3 See also
4 External links

[edit] Examples

The involute of a circle

Involute of a circle
The involute of a circle is a spiral. In cartesian coordinates the curve follows:

$x = a (cos(t) + t sin(t))$
$y = a (sin(t) - t cos(t))$
Where: t is the angle and a the radius

Involute of a catenary
The involute of a catenary through its vertex is a tractrix. In cartesian coordinates the curve follows:

$x = t - tanh(t)$
$y = sech(t)$
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:

$x = a (t + sin(t))$
$y = a (3 + cos(t))$
Where: t is the angle and a the radius

[edit] Application

The involute of a circle has a property that makes it important to the gear industry: if the teeth of two mating gears have the shape of an involute, their relative rates of rotation are constant while the teeth are engaged. With teeth of other shapes, the relative speeds 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.